Spring Final Exam Practice

1. Solve by using the Zero Product Property.

open parentheses 3 x minus 1 close parentheses open parentheses x plus 6 close parentheses equals 0

x = _____, _____

The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. In this case, the given answer of 1/3, -6, 1/3, -6 satisfies the Zero Product Property because when you multiply 1/3 and -6, you get zero. Therefore, the answer is valid.

Explanation

The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. In this case, the given answer of 1/3, -6, 1/3, -6 satisfies the Zero Product Property because when you multiply 1/3 and -6, you get zero. Therefore, the answer is valid.

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2. Solve by factoring.

x squared plus 4 x minus 45 equals 0

x = _____, _____

The given answer suggests that the equation can be solved by factoring. The equation is not provided, but it can be inferred that the equation has two solutions: -9 and 5. The repetition of -9 and 5 in the answer could indicate that these are the two solutions, possibly repeated for emphasis. However, without the actual equation, it is not possible to provide a definitive explanation.

Explanation

The given answer suggests that the equation can be solved by factoring. The equation is not provided, but it can be inferred that the equation has two solutions: -9 and 5. The repetition of -9 and 5 in the answer could indicate that these are the two solutions, possibly repeated for emphasis. However, without the actual equation, it is not possible to provide a definitive explanation.

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4. Solve by factoring.

16 plus x squared equals 8 x

x = _____

The equation "x = 4" can be solved by factoring. In this case, there is no need to factor because the equation is already in its simplest form. Therefore, the value of x is 4.

Explanation

The equation "x = 4" can be solved by factoring. In this case, there is no need to factor because the equation is already in its simplest form. Therefore, the value of x is 4.

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5. Solve by factoring.

14 x minus 49 equals x squared

x = _____

The equation x = 7 can be solved by factoring. By factoring out the common factor of 1, we are left with x = 7. This means that the value of x that satisfies the equation is 7.

Explanation

The equation x = 7 can be solved by factoring. By factoring out the common factor of 1, we are left with x = 7. This means that the value of x that satisfies the equation is 7.

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6. Solve by factoring.

4 x squared plus 4 x equals 15

x = _____, _____

The given answer suggests that the equation can be solved by factoring, resulting in two solutions: 3/2 and -5/2. These values are repeated twice, indicating that they are the correct solutions to the equation.

Explanation

The given answer suggests that the equation can be solved by factoring, resulting in two solutions: 3/2 and -5/2. These values are repeated twice, indicating that they are the correct solutions to the equation.

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7. Solve by factoring.

x squared plus 6 x minus 40 equals 0

x = _____, _____

The given answer suggests that the equation can be solved by factoring. By factoring the equation, we can find the values of x that satisfy the equation. The solution is x = -10 and x = 4. The answer is repeated twice, indicating that there are two solutions for x, which are -10 and 4.

Explanation

The given answer suggests that the equation can be solved by factoring. By factoring the equation, we can find the values of x that satisfy the equation. The solution is x = -10 and x = 4. The answer is repeated twice, indicating that there are two solutions for x, which are -10 and 4.

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9. Solve by factoring.

9 x squared minus 16 equals 0

x = _____, _____

The given answer is 4/3, -4/3, 4/3, -4/3. This means that the equation can be solved by factoring and the solutions are 4/3 and -4/3. The repetition of the solutions indicates that they are repeated roots.

Explanation

The given answer is 4/3, -4/3, 4/3, -4/3. This means that the equation can be solved by factoring and the solutions are 4/3 and -4/3. The repetition of the solutions indicates that they are repeated roots.

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10. Solve by factoring.

3 x squared plus 21 x minus 24 equals 0

x = _____, _____

The given equation can be solved by factoring. The factors of the equation are -8 and 1. Therefore, the values of x that satisfy the equation are -8 and 1. The repetition of -8 and 1 in the answer indicates that they are the repeated roots of the equation.

Explanation

The given equation can be solved by factoring. The factors of the equation are -8 and 1. Therefore, the values of x that satisfy the equation are -8 and 1. The repetition of -8 and 1 in the answer indicates that they are the repeated roots of the equation.

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11. Solve by factoring.

4 x squared minus 24 x minus 28 equals 0

x = _____, _____

The given answer suggests that the equation can be solved by factoring. The equation is x = 7, -1,7, -1. This means that the value of x can be either 7 or -1, and these values are repeated twice. Therefore, the solution to the equation is x = 7, -1.

Explanation

The given answer suggests that the equation can be solved by factoring. The equation is x = 7, -1,7, -1. This means that the value of x can be either 7 or -1, and these values are repeated twice. Therefore, the solution to the equation is x = 7, -1.

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12. Solve by factoring.

6 x squared plus 12 x equals 90

x = _____, _____

The equation can be solved by factoring. The factors of the equation are -5 and 3. Thus, the correct answer is -5, 3. The repetition of the answer (-5, 3, -5, 3) might be a typo or redundancy in the given options.

Explanation

The equation can be solved by factoring. The factors of the equation are -5 and 3. Thus, the correct answer is -5, 3. The repetition of the answer (-5, 3, -5, 3) might be a typo or redundancy in the given options.

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13. Solve by factoring.

6 x squared plus 18 x plus 12 equals 0

x = _____, _____

The given answer -2, -1, -2, -1 suggests that the equation can be solved by factoring. When factoring, we look for values that can be multiplied together to give us the desired result. In this case, the equation can be factored as (x + 2)(x + 1) = 0. By setting each factor equal to zero, we find that x = -2 and x = -1 are the solutions to the equation. The repetition of -2, -1 in the answer might be a typographical error or redundancy.

Explanation

The given answer -2, -1, -2, -1 suggests that the equation can be solved by factoring. When factoring, we look for values that can be multiplied together to give us the desired result. In this case, the equation can be factored as (x + 2)(x + 1) = 0. By setting each factor equal to zero, we find that x = -2 and x = -1 are the solutions to the equation. The repetition of -2, -1 in the answer might be a typographical error or redundancy.

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15. Solve by factoring.

42 equals 3 x squared plus 15 x

x = _____, _____

The given equation is solved by factoring. The factors of the equation are -7 and 2. The repeated factors indicate that the equation has a repeated root. Therefore, the solution to the equation is -7 and 2, with -7 being repeated twice.

Explanation

The given equation is solved by factoring. The factors of the equation are -7 and 2. The repeated factors indicate that the equation has a repeated root. Therefore, the solution to the equation is -7 and 2, with -7 being repeated twice.

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16. Solve by factoring.

72 plus 2 x squared equals 24 x

x = _____

To solve the equation by factoring, we need to find the value of x that makes the equation true. In this case, the equation is x = 6. By substituting 6 for x in the equation, we can see that it satisfies the equation and makes it true. Therefore, the correct answer is 6.

Explanation

To solve the equation by factoring, we need to find the value of x that makes the equation true. In this case, the equation is x = 6. By substituting 6 for x in the equation, we can see that it satisfies the equation and makes it true. Therefore, the correct answer is 6.

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17. Solve by factoring.

4 x minus 2 equals 2 x squared

x = _____

The given equation "x = 1" can be solved by factoring. However, since there is no expression or equation provided to factor, it is not possible to generate a solution or explanation for this question.

Explanation

The given equation "x = 1" can be solved by factoring. However, since there is no expression or equation provided to factor, it is not possible to generate a solution or explanation for this question.

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18. Solve by factoring.

7 x squared minus 28 equals 0

x = _____, _____

The given equation can be solved by factoring. The factors of the equation are (x - 2)(x + 2). Setting each factor equal to zero, we get x - 2 = 0 and x + 2 = 0. Solving these equations, we find x = 2 and x = -2. Therefore, the correct answer is 2, -2.

Explanation

The given equation can be solved by factoring. The factors of the equation are (x - 2)(x + 2). Setting each factor equal to zero, we get x - 2 = 0 and x + 2 = 0. Solving these equations, we find x = 2 and x = -2. Therefore, the correct answer is 2, -2.

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19. Solve by factoring.

3 x squared minus 48 equals 0

x = _____, _____

The given answer suggests that the equation can be solved by factoring. The equation is x = 4, -4, 4, -4. This means that when the equation is factored, it results in two sets of solutions: x = 4 and x = -4. Therefore, the possible values for x are 4 and -4.

Explanation

The given answer suggests that the equation can be solved by factoring. The equation is x = 4, -4, 4, -4. This means that when the equation is factored, it results in two sets of solutions: x = 4 and x = -4. Therefore, the possible values for x are 4 and -4.

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20. Solve by factoring.

0 equals x squared plus 11 x plus 30

x = _____, _____

The given expression is already factored, with the factors being -6 and -5. Therefore, the solution is x = -6 and x = -5.

Explanation

The given expression is already factored, with the factors being -6 and -5. Therefore, the solution is x = -6 and x = -5.

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22. Tell whether the set of events is independent or dependent.Pick "Joe" from a box of names, replace it, and then pick "Craig."

Independent

Dependent

The set of events is independent because picking "Joe" from the box and replacing it does not affect the probability of picking "Craig" afterwards. The outcome of the first event does not impact the outcome of the second event.

Explanation

The set of events is independent because picking "Joe" from the box and replacing it does not affect the probability of picking "Craig" afterwards. The outcome of the first event does not impact the outcome of the second event.

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24. Solve using the Quadratic Formula.

2 x squared plus 3 x equals 9

x = _____, x = _____

The given answer suggests that the quadratic equation has two distinct solutions, which are 3/2, -3, and 1.5. However, the answer is repeated twice, so it seems to be a duplication error. The correct answer should be 3/2, -3, and 1.5 only once.

Explanation

The given answer suggests that the quadratic equation has two distinct solutions, which are 3/2, -3, and 1.5. However, the answer is repeated twice, so it seems to be a duplication error. The correct answer should be 3/2, -3, and 1.5 only once.

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26. A spinner has an equal chance of landing on 1 of 3 colors: purple, blue, or green. What is the probability of landing on purple on two consecutive spins?Express your answer as a fraction using the [slash] key for a fraction bar. For example,

2 over 15

should be entered as 2/15.

The probability of landing on purple on the first spin is 1/3. Since each spin is independent, the probability of landing on purple on the second spin is also 1/3. To find the probability of both events happening, we multiply the probabilities together: (1/3) * (1/3) = 1/9. Therefore, the probability of landing on purple on two consecutive spins is 1/9.

Explanation

The probability of landing on purple on the first spin is 1/3. Since each spin is independent, the probability of landing on purple on the second spin is also 1/3. To find the probability of both events happening, we multiply the probabilities together: (1/3) * (1/3) = 1/9. Therefore, the probability of landing on purple on two consecutive spins is 1/9.

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27. Find the probability if you spin the spinner and roll the number cube.P(striped, even)Express your answer as a fraction using the [slash] key for a fraction bar. For example,

2 over 15

should be entered as 2/15.

The probability of spinning a striped section on the spinner is 1 out of 8 possible outcomes. The probability of rolling an even number on the number cube is also 1 out of 8 possible outcomes. Since both events are independent, we can multiply their probabilities together to find the probability of both events happening. Therefore, the probability of spinning a striped section and rolling an even number is 1/8 multiplied by 1/8, which simplifies to 1/64.

Explanation

The probability of spinning a striped section on the spinner is 1 out of 8 possible outcomes. The probability of rolling an even number on the number cube is also 1 out of 8 possible outcomes. Since both events are independent, we can multiply their probabilities together to find the probability of both events happening. Therefore, the probability of spinning a striped section and rolling an even number is 1/8 multiplied by 1/8, which simplifies to 1/64.

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28. What is the value of the discriminant of the equation

8 x squared plus 26 x plus 15 equals 0

?

The value of the discriminant of an equation is used to determine the nature of its solutions. In this case, the given value of 196 is the discriminant itself. Since the discriminant is positive (equal to 196), it indicates that the equation has two distinct real solutions.

Explanation

The value of the discriminant of an equation is used to determine the nature of its solutions. In this case, the given value of 196 is the discriminant itself. Since the discriminant is positive (equal to 196), it indicates that the equation has two distinct real solutions.

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29. Find the probability if you spin the spinner and roll the number cube.P(not white, less than 5)Express your answer as a fraction using the [slash] key for a fraction bar. For example,

2 over 15

should be entered as 2/15.

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Explanation

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30. What is the value of the discriminant of the equation

x squared plus 11 x minus 20 equals 0

?

The value of the discriminant of an equation is found by using the formula b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. In this case, since there is no equation given, it is not possible to calculate the discriminant. Therefore, an explanation for the given answer is not available.

Explanation

The value of the discriminant of an equation is found by using the formula b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. In this case, since there is no equation given, it is not possible to calculate the discriminant. Therefore, an explanation for the given answer is not available.

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31. Find the probability if you spin the spinner and roll the number cube.P(white, 8)Express your answer as a fraction using the [slash] key for a fraction bar. For example,

2 over 15

should be entered as 2/15.

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Explanation

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Submit

32. What is the value of the discriminant of the equation

negative 2 x squared plus x minus 5 equals 0

?

The value of the discriminant of an equation is used to determine the nature of the roots of the equation. In this case, since the value of the discriminant is -39, it indicates that the equation has two complex conjugate roots.

Explanation

The value of the discriminant of an equation is used to determine the nature of the roots of the equation. In this case, since the value of the discriminant is -39, it indicates that the equation has two complex conjugate roots.

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33. A bag contains 2 red marbles, 3 yellow marbles, and 4 green marbles. Find the probability if you pick two marbles without replacing the first.P(red, then green)Express your answer as a fraction using the [slash] key for a fraction bar. For example,

2 over 15

should be entered as 2/15.

The probability of picking a red marble on the first draw is 2/9. Since the first marble is not replaced, there are now 8 marbles left in the bag, with 4 of them being green. Therefore, the probability of picking a green marble on the second draw is 4/8, which simplifies to 1/2. To find the probability of both events happening, we multiply the probabilities: (2/9) * (1/2) = 2/18, which simplifies to 1/9.

Explanation

The probability of picking a red marble on the first draw is 2/9. Since the first marble is not replaced, there are now 8 marbles left in the bag, with 4 of them being green. Therefore, the probability of picking a green marble on the second draw is 4/8, which simplifies to 1/2. To find the probability of both events happening, we multiply the probabilities: (2/9) * (1/2) = 2/18, which simplifies to 1/9.

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34. What is the value of the discriminant of the equation

3 x squared plus 5 x minus 16 equals 0

?

The value of the discriminant of an equation is equal to the square of the coefficient of the x-term minus 4 times the product of the coefficients of the x-term and the constant term. In this case, the equation is not given, only the value of the discriminant, which is 217. Therefore, we cannot determine the equation or the coefficients from the given information.

Explanation

The value of the discriminant of an equation is equal to the square of the coefficient of the x-term minus 4 times the product of the coefficients of the x-term and the constant term. In this case, the equation is not given, only the value of the discriminant, which is 217. Therefore, we cannot determine the equation or the coefficients from the given information.

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35. A bag contains 2 red marbles, 3 yellow marbles, and 4 green marbles. Find the probability if you pick two marbles without replacing the first.P(red, then yellow)Express your answer as a fraction using the [slash] key for a fraction bar. For example,

2 over 15

should be entered as 2/15.

The probability of picking a red marble first is 2/9, since there are 2 red marbles out of a total of 9 marbles. After removing one red marble, there are now 8 marbles left in the bag, with 3 of them being yellow. Therefore, the probability of picking a yellow marble second is 3/8. To find the probability of both events happening, we multiply the individual probabilities together: (2/9) * (3/8) = 6/72 = 1/12.

Explanation

The probability of picking a red marble first is 2/9, since there are 2 red marbles out of a total of 9 marbles. After removing one red marble, there are now 8 marbles left in the bag, with 3 of them being yellow. Therefore, the probability of picking a yellow marble second is 3/8. To find the probability of both events happening, we multiply the individual probabilities together: (2/9) * (3/8) = 6/72 = 1/12.

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36. What is the value of the discriminant of the equation

12 x squared plus 4 x minus 9 equals 0

?

The value of the discriminant of an equation is a mathematical term used to determine the nature of the solutions. It is calculated by taking the square root of the expression inside the quadratic formula. In this case, the value of the discriminant is given as 448. Since the discriminant is positive, it means that the equation has two distinct real solutions.

Explanation

The value of the discriminant of an equation is a mathematical term used to determine the nature of the solutions. It is calculated by taking the square root of the expression inside the quadratic formula. In this case, the value of the discriminant is given as 448. Since the discriminant is positive, it means that the equation has two distinct real solutions.

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37. A bag contains 2 red marbles, 3 yellow marbles, and 4 green marbles. Find the probability if you pick two marbles without replacing the first.P(green, then green)Express your answer as a fraction using the [slash] key for a fraction bar. For example,

2 over 15

should be entered as 2/15.

The probability of picking a green marble on the first draw is 4/9, since there are 4 green marbles out of a total of 9 marbles. After removing the first green marble, there are now 8 marbles left, with 3 green marbles remaining. Therefore, the probability of picking a green marble on the second draw, without replacement, is 3/8. To find the probability of both events happening, we multiply the probabilities together: (4/9) * (3/8) = 12/72 = 1/6.

Explanation

The probability of picking a green marble on the first draw is 4/9, since there are 4 green marbles out of a total of 9 marbles. After removing the first green marble, there are now 8 marbles left, with 3 green marbles remaining. Therefore, the probability of picking a green marble on the second draw, without replacement, is 3/8. To find the probability of both events happening, we multiply the probabilities together: (4/9) * (3/8) = 12/72 = 1/6.

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38. What is the value of the discriminant of the equation

1 half x squared minus 6 x plus 18 equals 0

?

The value of the discriminant of an equation is determined by the coefficients of the equation. In this case, since the value of the discriminant is given as 0, it means that the equation has only one real root. This occurs when the equation has a perfect square trinomial or when the equation has repeated roots.

Explanation

The value of the discriminant of an equation is determined by the coefficients of the equation. In this case, since the value of the discriminant is given as 0, it means that the equation has only one real root. This occurs when the equation has a perfect square trinomial or when the equation has repeated roots.

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39. A bag contains 2 red marbles, 3 yellow marbles, and 4 green marbles. Find the probability if you pick two marbles without replacing the first.P(red, then red)Express your answer as a fraction using the [slash] key for a fraction bar. For example,

2 over 15

should be entered as 2/15.

The probability of picking a red marble on the first draw is 2/9 since there are 2 red marbles out of a total of 9 marbles. After the first marble is picked without replacement, there is 1 red marble left out of a total of 8 marbles. Therefore, the probability of picking a red marble on the second draw is 1/8. To find the probability of both events happening, we multiply the probabilities together: (2/9) * (1/8) = 1/36.

Explanation

The probability of picking a red marble on the first draw is 2/9 since there are 2 red marbles out of a total of 9 marbles. After the first marble is picked without replacement, there is 1 red marble left out of a total of 8 marbles. Therefore, the probability of picking a red marble on the second draw is 1/8. To find the probability of both events happening, we multiply the probabilities together: (2/9) * (1/8) = 1/36.

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40. What is the value of the discriminant of the equation

3 x squared plus 5 x plus 4 equals 0

?

The value of the discriminant of an equation is determined by the coefficients of the quadratic equation and is calculated as b^2 - 4ac. In this case, since the value of the discriminant is given as -23, it means that b^2 - 4ac is equal to -23.

Explanation

The value of the discriminant of an equation is determined by the coefficients of the quadratic equation and is calculated as b^2 - 4ac. In this case, since the value of the discriminant is given as -23, it means that b^2 - 4ac is equal to -23.

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41. Use a calculator to find a decimal approximation (to the nearest hundredth) for the solution

x equals fraction numerator negative 1 plus-or-minus square root of 57 over denominator 4 end fraction

x almost equal to

_____, _____

The given answer repeats the values -2.14 and 1.64 twice. This suggests that the decimal approximation (to the nearest hundredth) for the solution is -2.14 and 1.64.

Explanation

The given answer repeats the values -2.14 and 1.64 twice. This suggests that the decimal approximation (to the nearest hundredth) for the solution is -2.14 and 1.64.

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43. An experiment consists of randomly choosing a marble from a bag. Use the results in the table to find the experimental probability of each event.
Not choosing a green marble.

To find the experimental probability of not choosing a green marble, we need to calculate the number of times a non-green marble was chosen and divide it by the total number of trials. Looking at the table, we can see that there are a total of 40 trials and 30 of them resulted in choosing a non-green marble. Therefore, the experimental probability of not choosing a green marble is 30/40, which simplifies to 3/4 or 0.75.

Explanation

To find the experimental probability of not choosing a green marble, we need to calculate the number of times a non-green marble was chosen and divide it by the total number of trials. Looking at the table, we can see that there are a total of 40 trials and 30 of them resulted in choosing a non-green marble. Therefore, the experimental probability of not choosing a green marble is 30/40, which simplifies to 3/4 or 0.75.

44. What does it mean for a quadratic function to have no zeros?

The graph does not touch or pass through the x-axis

The function is not made up of a factorable rule

The function is undefined when x=0

The graph is not a parabola

When a quadratic function has no zeros, it means that the graph of the function does not intersect or touch the x-axis. In other words, there are no values of x for which the function equals zero. This can be visualized as a parabola that does not cross the x-axis and remains either entirely above or entirely below it. The other options provided in the question are not accurate explanations for a quadratic function having no zeros.

Explanation

When a quadratic function has no zeros, it means that the graph of the function does not intersect or touch the x-axis. In other words, there are no values of x for which the function equals zero. This can be visualized as a parabola that does not cross the x-axis and remains either entirely above or entirely below it. The other options provided in the question are not accurate explanations for a quadratic function having no zeros.

45. Find the axis of symmetry of the parabola with zeros -1 and 6

X = 1.5

X = 2

X = 2.5

X = 3

The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. The vertex is the midpoint between the zeros of the parabola. In this case, the zeros are -1 and 6. The midpoint between -1 and 6 is (6 - 1) / 2 = 2.5. Therefore, the axis of symmetry of the parabola is x = 2.5.

Explanation

The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. The vertex is the midpoint between the zeros of the parabola. In this case, the zeros are -1 and 6. The midpoint between -1 and 6 is (6 - 1) / 2 = 2.5. Therefore, the axis of symmetry of the parabola is x = 2.5.

46. What is the formula for finding the axis of symmetry?

The formula for finding the axis of symmetry of a quadratic function is x = -b/2a, where a, b, and c are coefficients of the quadratic equation in the form ax^2 + bx + c. This formula is derived from completing the square method and represents the x-coordinate of the vertex of the parabola.

Explanation

The formula for finding the axis of symmetry of a quadratic function is x = -b/2a, where a, b, and c are coefficients of the quadratic equation in the form ax^2 + bx + c. This formula is derived from completing the square method and represents the x-coordinate of the vertex of the parabola.

47. Identify the vertex of the given parabola.

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Explanation

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48. Because a parabola is symmetrical, each point is the same number of units away from the axis of symmetry as its reflected point.

True

False

A parabola is a symmetrical curve that has an axis of symmetry. This means that if you take any point on the parabola and draw a line perpendicular to the axis of symmetry, the distance from that point to the axis of symmetry will be the same as the distance from its reflected point on the other side of the axis. Therefore, the statement "Because a parabola is symmetrical, each point is the same number of units away from the axis of symmetry as its reflected point" is true.

Explanation

A parabola is a symmetrical curve that has an axis of symmetry. This means that if you take any point on the parabola and draw a line perpendicular to the axis of symmetry, the distance from that point to the axis of symmetry will be the same as the distance from its reflected point on the other side of the axis. Therefore, the statement "Because a parabola is symmetrical, each point is the same number of units away from the axis of symmetry as its reflected point" is true.

49. Solve by using the Zero Product Property.

open parentheses 5 x plus 3 close parentheses open parentheses x minus 4 close parentheses equals 0

x = _____, _____

The given answer -3/5, 4,-3/5, 4 is correct. The Zero Product Property states that if the product of two or more numbers is zero, then at least one of the numbers must be zero. In this case, the two numbers -3/5 and 4 are multiplied together to give a product of zero. Therefore, either -3/5 or 4 (or both) must be zero. This explains why the answer includes both -3/5 and 4 twice.

Explanation

The given answer -3/5, 4,-3/5, 4 is correct. The Zero Product Property states that if the product of two or more numbers is zero, then at least one of the numbers must be zero. In this case, the two numbers -3/5 and 4 are multiplied together to give a product of zero. Therefore, either -3/5 or 4 (or both) must be zero. This explains why the answer includes both -3/5 and 4 twice.

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50. Solve by factoring.

7 minus 5 x squared equals 2 x

x = _____, _____

The given equation can be solved by factoring. The factors of the equation are -7/5 and 1. Therefore, the solution to the equation is -7/5 and 1.

Explanation

The given equation can be solved by factoring. The factors of the equation are -7/5 and 1. Therefore, the solution to the equation is -7/5 and 1.

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51. Solve by factoring.

2 x squared plus 6 x equals 0

x = _____, _____

The equation can be solved by factoring. By setting each factor equal to zero, we find that x can be either 0 or -3. Therefore, the correct answer is 0, -3, 0, -3.

Explanation

The equation can be solved by factoring. By setting each factor equal to zero, we find that x can be either 0 or -3. Therefore, the correct answer is 0, -3, 0, -3.

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52. Factor the polynomial completely.

9 p squared minus q squared plus 3 p

Unfactorable

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Explanation

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53. An experiment consists of randomly choosing a marble from a bag. Use the results in the table to find the experimental probability of each event.Choosing a yellow marble.

The experimental probability of choosing a yellow marble can be found by dividing the number of times a yellow marble was chosen (4) by the total number of marbles chosen (10). This gives us a probability of 4/10 or 2/5.

Explanation

The experimental probability of choosing a yellow marble can be found by dividing the number of times a yellow marble was chosen (4) by the total number of marbles chosen (10). This gives us a probability of 4/10 or 2/5.

54. Tell whether the following statement is SOMETIMES, ALWAYS, or NEVER true.The graph of a quadratic function is a straight line.

Sometimes true

Always true

Never true

The graph of a quadratic function is never a straight line because a quadratic function is a polynomial function of degree 2, which means it has a squared term. The graph of a quadratic function is always a curve, either concave up or concave down, and never a straight line.

Explanation

The graph of a quadratic function is never a straight line because a quadratic function is a polynomial function of degree 2, which means it has a squared term. The graph of a quadratic function is always a curve, either concave up or concave down, and never a straight line.

55. Which of the following is true of a parabola?

It is the shape of the graph of a quadratic function.

It has a low point (minimum) or a high point (maximum).

It can be divided into two identical, though mirror-image, halves.

All of the above

A parabola is the shape of the graph of a quadratic function. It can have a low point (minimum) or a high point (maximum), depending on the direction it opens. Additionally, a parabola can be divided into two identical, though mirror-image, halves. Therefore, all of the given statements are true of a parabola.

Explanation

A parabola is the shape of the graph of a quadratic function. It can have a low point (minimum) or a high point (maximum), depending on the direction it opens. Additionally, a parabola can be divided into two identical, though mirror-image, halves. Therefore, all of the given statements are true of a parabola.

56. Find the theoretical probability of randomly choosing a blue marble from a bag of 5 blue marbles, 8 red marbles, and 7 yellow marbles.

25%

50%

75%

The theoretical probability of randomly choosing a blue marble can be found by dividing the number of blue marbles by the total number of marbles in the bag. In this case, there are 5 blue marbles out of a total of 20 marbles (5 blue + 8 red + 7 yellow). Therefore, the probability is 5/20, which simplifies to 1/4 or 25%.

Explanation

The theoretical probability of randomly choosing a blue marble can be found by dividing the number of blue marbles by the total number of marbles in the bag. In this case, there are 5 blue marbles out of a total of 20 marbles (5 blue + 8 red + 7 yellow). Therefore, the probability is 5/20, which simplifies to 1/4 or 25%.

57. The vertex of this parabola shows that the __________ value of the function is _____.

Minimum; -5

Maximum; 1

Minimum; -9

Maximum; 2

The vertex of a parabola represents the highest or lowest point on the graph, depending on the direction of the curve. In this case, the vertex represents the minimum value of the function. The y-coordinate of the vertex is -9, indicating that the function reaches its lowest point at that value.

Explanation

The vertex of a parabola represents the highest or lowest point on the graph, depending on the direction of the curve. In this case, the vertex represents the minimum value of the function. The y-coordinate of the vertex is -9, indicating that the function reaches its lowest point at that value.

58. Find the vertex of the graph of the function

y equals negative 4 x squared minus 8 x plus 5

.

(–1, 9)

(4, –91)

(1, –7)

(–2, 5)

The vertex of a graph represents the highest or lowest point on the graph. In this case, the vertex is given as (-1, 9), which means that the graph reaches its highest point at x = -1 and y = 9.

Explanation

The vertex of a graph represents the highest or lowest point on the graph. In this case, the vertex is given as (-1, 9), which means that the graph reaches its highest point at x = -1 and y = 9.

59. What are the x-intercepts of the graph of the function

f open parentheses x close parentheses equals open parentheses x plus 3 close parentheses open parentheses x minus 4 close parentheses

?

3 and 4

–3 and –4

3 and –4

–3 and 4

The x-intercepts of a graph are the points where the graph intersects the x-axis. In this case, the x-intercepts are -3 and 4. This means that when the function is graphed, the points (-3, 0) and (4, 0) will be on the graph, indicating that the function crosses the x-axis at these points.

Explanation

The x-intercepts of a graph are the points where the graph intersects the x-axis. In this case, the x-intercepts are -3 and 4. This means that when the function is graphed, the points (-3, 0) and (4, 0) will be on the graph, indicating that the function crosses the x-axis at these points.

60. Complete the square for the given expression. Write the resulting expression as a binomial squared.

x squared minus 18 x plus box enclose space space space space space space space space end enclose

The correct answer is not here

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Explanation

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61. Solve by using the Zero Product Property.

open parentheses 3 x minus 4 close parentheses open parentheses 2 x plus 1 close parentheses equals 0

x = _____, _____

The given answer is incorrect. The correct answer should be 4/3, -1/2. The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. In this case, the equation x = 4/3, -1/2 would satisfy the Zero Product Property because when we substitute these values into the equation, we get (4/3)(-1/2) = 0. Therefore, the correct answer is 4/3, -1/2.

Explanation

The given answer is incorrect. The correct answer should be 4/3, -1/2. The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. In this case, the equation x = 4/3, -1/2 would satisfy the Zero Product Property because when we substitute these values into the equation, we get (4/3)(-1/2) = 0. Therefore, the correct answer is 4/3, -1/2.

Submit

62. Solve using the Quadratic Formula.

3 x squared minus 2 x equals 8

x = _____, x = _____

The quadratic formula is used to find the solutions to a quadratic equation in the form of ax^2 + bx + c = 0. In this case, the solutions are x = -4/3 and x = 2. This means that when the quadratic equation is solved, it will result in these two values for x. The repeated appearance of -4/3 and 2 in the answer may be a formatting error or redundancy.

Explanation

The quadratic formula is used to find the solutions to a quadratic equation in the form of ax^2 + bx + c = 0. In this case, the solutions are x = -4/3 and x = 2. This means that when the quadratic equation is solved, it will result in these two values for x. The repeated appearance of -4/3 and 2 in the answer may be a formatting error or redundancy.

Submit

63. What is the definition of the axis of symmetry?

The value of x when y=0

Highest power of the independent variable

The maximum or minimum of the function

A vertical line that divides a parabola into two symmetrical halves

The axis of symmetry is a vertical line that divides a parabola into two symmetrical halves. It is not the value of x when y=0, the highest power of the independent variable, or the maximum or minimum of the function.

Explanation

The axis of symmetry is a vertical line that divides a parabola into two symmetrical halves. It is not the value of x when y=0, the highest power of the independent variable, or the maximum or minimum of the function.

64. Find the axis of symmetry of the graph of the function

y equals 3 x squared minus 24 x minus 27

.

X = 3

X = 6

X = –2

X = 4

The axis of symmetry of a graph is a vertical line that divides the graph into two symmetric parts. In this case, the given answer x = 4 represents the axis of symmetry. This means that the graph of the function will be symmetric with respect to the line x = 4.

Explanation

The axis of symmetry of a graph is a vertical line that divides the graph into two symmetric parts. In this case, the given answer x = 4 represents the axis of symmetry. This means that the graph of the function will be symmetric with respect to the line x = 4.

65. Solve by factoring.

5 x squared minus 125 equals 0

x = _____, _____

The equation can be solved by factoring. The factors of the equation are 5 and -5. When multiplied together, they give a product of 25. Therefore, the values of x that satisfy the equation are 5 and -5.

Explanation

The equation can be solved by factoring. The factors of the equation are 5 and -5. When multiplied together, they give a product of 25. Therefore, the values of x that satisfy the equation are 5 and -5.

Submit

66. Find the axis of symmetry of the parabola with zeros

negative 4 space text and   end text 2

.

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Explanation

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67. Find the axis of symmetry of the graph of the function

y equals x squared plus 4 x minus 7

.

X = 1

X = 2

X = –2

X = 4

The correct answer is x = –2 because the axis of symmetry is a vertical line that divides the graph into two equal halves. In this case, the line x = –2 will divide the graph of the function into two equal halves, with the same values of y on either side of the line.

Explanation

The correct answer is x = –2 because the axis of symmetry is a vertical line that divides the graph into two equal halves. In this case, the line x = –2 will divide the graph of the function into two equal halves, with the same values of y on either side of the line.

68. Solve by factoring.

3 x squared plus x equals 0

x = _____, _____

The equation is solved by factoring. The factors of the equation are 0 and -1/3. Therefore, the solution to the equation is x = 0 and x = -1/3.

Explanation

The equation is solved by factoring. The factors of the equation are 0 and -1/3. Therefore, the solution to the equation is x = 0 and x = -1/3.

Submit

70. An experiment consists of randomly choosing a marble from a bag. Use the results in the table to find the experimental probability of each event.
Choosing a blue marble.

The experimental probability of choosing a blue marble can be found by dividing the number of times a blue marble was chosen (4) by the total number of trials (10). This gives us an experimental probability of 4/10 or 0.4.

Explanation

The experimental probability of choosing a blue marble can be found by dividing the number of times a blue marble was chosen (4) by the total number of trials (10). This gives us an experimental probability of 4/10 or 0.4.

71. Use the spinner to find the following.P(green)

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Explanation

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72. Given the standard form of a quadratic function,

y equals a x squared plus b x plus c

, which of the following indicates a parabola that opens downward?

The standard form of a quadratic function is y = ax^2 + bx + c. In this form, the coefficient 'a' determines the direction of the parabola. If 'a' is negative, the parabola opens downward. Therefore, the correct answer would be the option that has a negative coefficient for 'a'.

Explanation

The standard form of a quadratic function is y = ax^2 + bx + c. In this form, the coefficient 'a' determines the direction of the parabola. If 'a' is negative, the parabola opens downward. Therefore, the correct answer would be the option that has a negative coefficient for 'a'.

73. Use the spinner to find the following.
P(not blue)

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Explanation

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74. Solve by factoring.

18 x minus 24 equals 3 x squared

x = _____, _____

The given answer, 2, 4, 2, 4, suggests that the equation can be solved by factoring. Factoring involves breaking down the equation into its factors to find the values of x. In this case, the factors of the equation are 2 and 4, and since they are repeated twice, they are both solutions to the equation. Therefore, the values of x are 2 and 4.

Explanation

The given answer, 2, 4, 2, 4, suggests that the equation can be solved by factoring. Factoring involves breaking down the equation into its factors to find the values of x. In this case, the factors of the equation are 2 and 4, and since they are repeated twice, they are both solutions to the equation. Therefore, the values of x are 2 and 4.

Submit

75. Solve by factoring.

x plus 56 equals x squared

x = _____, _____

The given answer suggests that the equation can be solved by factoring and the solutions are -7 and 8. It seems that the equation has a repeated root, hence the repetition of -7 and 8 in the answer. However, without the complete question or context, it is difficult to provide a more accurate explanation.

Explanation

The given answer suggests that the equation can be solved by factoring and the solutions are -7 and 8. It seems that the equation has a repeated root, hence the repetition of -7 and 8 in the answer. However, without the complete question or context, it is difficult to provide a more accurate explanation.

Submit

77. The vertex of this parabola shows that the __________ value of the function is _____.

Minimum; 2

Maximum; 4

Minimum; 4

Maximum; 2

The vertex of a parabola represents the highest or lowest point on the graph. In this case, the vertex is at its highest point, indicating a maximum value. The y-coordinate of the vertex is 4, which means that the maximum value of the function is 4.

Explanation

The vertex of a parabola represents the highest or lowest point on the graph. In this case, the vertex is at its highest point, indicating a maximum value. The y-coordinate of the vertex is 4, which means that the maximum value of the function is 4.

78. Solve the equation.

x squared minus 6 x plus 9 equals 100

3 and 10

–10 and –3

–7 and 13

The correct solution is not here

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Explanation

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79. Tell whether the following statement is SOMETIMES, ALWAYS, or NEVER true.The highest power of the independent variable in a quadratic function is 2.

Sometimes true

Always true

Never true

In a quadratic function, the highest power of the independent variable is always 2. This is because a quadratic function is defined as a function of the form f(x) = ax^2 + bx + c, where the highest power of x is 2. Therefore, it is always true that the highest power of the independent variable in a quadratic function is 2.

Explanation

In a quadratic function, the highest power of the independent variable is always 2. This is because a quadratic function is defined as a function of the form f(x) = ax^2 + bx + c, where the highest power of x is 2. Therefore, it is always true that the highest power of the independent variable in a quadratic function is 2.

80. Which of the following is the equivalent of a zero of a function?

The vertex

The axis of symmetry

The x-intercept

The y-intercept

The x-intercept of a function is the point where the graph of the function intersects the x-axis. At this point, the value of the function is equal to zero. Therefore, the x-intercept is equivalent to a zero of a function.

Explanation

The x-intercept of a function is the point where the graph of the function intersects the x-axis. At this point, the value of the function is equal to zero. Therefore, the x-intercept is equivalent to a zero of a function.

81. Complete the square for the given expression. Write the resulting expression as a binomial squared.

x squared plus 10 x plus box enclose space space space space space end enclose

The correct answer is not here

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Explanation

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82. Solve by factoring.

2 x squared minus 3 x equals 9

x = _____, _____

The given equation can be solved by factoring. The factors of the equation are -3/2 and 3. Therefore, the solution to the equation is -3/2 and 3.

Explanation

The given equation can be solved by factoring. The factors of the equation are -3/2 and 3. Therefore, the solution to the equation is -3/2 and 3.

Submit

83. Which of the following is true of the graph of the function

f open parentheses x close parentheses equals x squared plus 3 x plus 5

?

The axis of symmetry is x = 1.5.

The y-intercept is +5.

The parabola opens downward.

The rule is factorable to (x+5)(x+1).

The y-intercept of a function is the point where the graph intersects the y-axis. In this case, the y-intercept is given as +5, which means that the graph intersects the y-axis at the point (0, 5). This information tells us that when x = 0, the value of the function is 5. Therefore, the statement "The y-intercept is +5" is true.

Explanation

The y-intercept of a function is the point where the graph intersects the y-axis. In this case, the y-intercept is given as +5, which means that the graph intersects the y-axis at the point (0, 5). This information tells us that when x = 0, the value of the function is 5. Therefore, the statement "The y-intercept is +5" is true.

84. Solve by factoring.

5 x squared minus 4 equals 8 x

x = _____, _____

The given answer states that the values of x are -2/5 and 2. However, it repeats the values twice, which seems to be a duplication error. The correct answer should be -2/5 and 2, without any repetition.

Explanation

The given answer states that the values of x are -2/5 and 2. However, it repeats the values twice, which seems to be a duplication error. The correct answer should be -2/5 and 2, without any repetition.

Submit

85. Solve using the Quadratic Formula.

x squared plus 3 x minus 8 equals 0

The quadratic formula is a method used to find the solutions to a quadratic equation of the form ax^2 + bx + c = 0. It is derived by completing the square and provides a formula to calculate the values of x that satisfy the equation. By substituting the given values of a, b, and c into the quadratic formula, we can solve for the values of x.

Explanation

The quadratic formula is a method used to find the solutions to a quadratic equation of the form ax^2 + bx + c = 0. It is derived by completing the square and provides a formula to calculate the values of x that satisfy the equation. By substituting the given values of a, b, and c into the quadratic formula, we can solve for the values of x.

86. Factor the polynomial completely.

7 x to the power of 5 plus 21 x to the power of 4 minus 28 x cubed

Unfactorable

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Explanation

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87. Given the standard form of a quadratic function,

y equals a x squared plus b x plus c

, which of the following indicates a parabola that opens upward?

A quadratic function in standard form is written as f(x) = ax^2 + bx + c, where a, b, and c are constants. In this form, the coefficient of the x^2 term (a) determines the direction of the parabola. If a is positive, the parabola opens upward. Therefore, the correct answer would be the option that has a positive coefficient for the x^2 term.

Explanation

A quadratic function in standard form is written as f(x) = ax^2 + bx + c, where a, b, and c are constants. In this form, the coefficient of the x^2 term (a) determines the direction of the parabola. If a is positive, the parabola opens upward. Therefore, the correct answer would be the option that has a positive coefficient for the x^2 term.

88. Find the axis of symmetry of the parabola with zeros 2 and 12.

X = 6

X = 7

X = 10

X = 24

The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. In this case, since the zeros of the parabola are 2 and 12, the vertex lies exactly in the middle of these two zeros. Therefore, the x-coordinate of the vertex is the average of the two zeros, which is (2 + 12) / 2 = 7. Hence, the equation of the axis of symmetry is x = 7.

Explanation

The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. In this case, since the zeros of the parabola are 2 and 12, the vertex lies exactly in the middle of these two zeros. Therefore, the x-coordinate of the vertex is the average of the two zeros, which is (2 + 12) / 2 = 7. Hence, the equation of the axis of symmetry is x = 7.

89. Find the vertex of the graph of the function

y equals x squared plus 6 x minus 16

.

(6, 24)

(–3, –25)

(3, 11)

(–6, –16)

The vertex of a graph represents the highest or lowest point on the graph. In this case, the vertex is represented by the point (-3, -25), which means that the function reaches its lowest point at x = -3 and y = -25. This point is the lowest point on the graph and is also known as the minimum point.

Explanation

The vertex of a graph represents the highest or lowest point on the graph. In this case, the vertex is represented by the point (-3, -25), which means that the function reaches its lowest point at x = -3 and y = -25. This point is the lowest point on the graph and is also known as the minimum point.

90. Solve the equation by completing the square.

x squared minus 4 x equals negative 1

Correct solution is not here

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Explanation

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91. Tell whether the set of events is independent or dependent.Roll an even number on a number cube and get tails when tossing a coin.

Independent

Dependent

The set of events is independent because rolling an even number on a number cube does not affect the outcome of getting tails when tossing a coin. The two events are unrelated and the outcome of one event does not influence the outcome of the other event.

Explanation

The set of events is independent because rolling an even number on a number cube does not affect the outcome of getting tails when tossing a coin. The two events are unrelated and the outcome of one event does not influence the outcome of the other event.

92. Choose the function that best represents the graph.

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Explanation

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93. Solve using the Quadratic Formula.

x squared equals 4 x plus 7

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Explanation

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94. Solve by factoring.

5 equals 6 x squared plus 7 x

x = _____, _____

The given answer 1/2, -5/3, 1/2, -5/3 suggests that the equation can be solved by factoring. By factoring, we can find the values of x that satisfy the equation. The repeated values 1/2 and -5/3 indicate that these are the solutions for x.

Explanation

The given answer 1/2, -5/3, 1/2, -5/3 suggests that the equation can be solved by factoring. By factoring, we can find the values of x that satisfy the equation. The repeated values 1/2 and -5/3 indicate that these are the solutions for x.

Submit

95. Tell whether the set of events is independent or dependent.You draw a heart from a deck of cards and set it aside. Then you draw a club from the deck of cards.

Independent

Dependent

The events of drawing a heart and drawing a club are dependent because the outcome of the first event affects the probability of the second event. When a heart is drawn and set aside, there is one less heart in the deck, which changes the probability of drawing a club from the remaining cards. Therefore, the events are dependent.

Explanation

The events of drawing a heart and drawing a club are dependent because the outcome of the first event affects the probability of the second event. When a heart is drawn and set aside, there is one less heart in the deck, which changes the probability of drawing a club from the remaining cards. Therefore, the events are dependent.

96. Use a calculator to find a decimal approximation (to the nearest hundredth) for the solution

x equals fraction numerator negative 3 plus-or-minus square root of 37 over denominator 2 end fraction

x almost equal to

_____, _____

The given answer is a repetition of the options provided in the question. It seems like there is an error in the question or the answer choices have been repeated unintentionally. Without any additional information or context, it is not possible to determine the correct explanation for this question.

Explanation

The given answer is a repetition of the options provided in the question. It seems like there is an error in the question or the answer choices have been repeated unintentionally. Without any additional information or context, it is not possible to determine the correct explanation for this question.

Submit

97. Find the zeros of

y equals x squared minus 8 x plus 12

from its graph below.

2 and 6

-6 and -2

0 and 4

-4 and 4

The graph of the given equation intersects the x-axis at 2 and 6. These are the points where the value of the equation is equal to zero, indicating that these are the zeros of the equation.

Explanation

The graph of the given equation intersects the x-axis at 2 and 6. These are the points where the value of the equation is equal to zero, indicating that these are the zeros of the equation.

98. Factor the polynomial completely.

4 x cubed plus 8 x squared plus 4 x

Unfactorable

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Explanation

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99. What is the maximum number of zeros a quadratic function can have?

1

2

3

4

A quadratic function is defined by an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants. The maximum number of zeros a quadratic function can have is 2. This is because a quadratic function represents a parabola, which can intersect the x-axis at most twice. The zeros of the function correspond to the x-intercepts of the parabola. Therefore, the correct answer is 2.

Explanation

A quadratic function is defined by an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants. The maximum number of zeros a quadratic function can have is 2. This is because a quadratic function represents a parabola, which can intersect the x-axis at most twice. The zeros of the function correspond to the x-intercepts of the parabola. Therefore, the correct answer is 2.

100. A bag contains 5 red marbles, 3 white marbles, and 4 blue marbles. What is the probability of randomly selecting a red marble, setting it aside, and then randomly selecting another red marble from the bag?

The probability of selecting a red marble and setting it aside is 5/12, as there are 5 red marbles out of a total of 12 marbles. After setting aside one red marble, there are now 4 red marbles left in the bag. Therefore, the probability of randomly selecting another red marble is 4/11. To find the overall probability, we multiply these two probabilities together: (5/12) * (4/11) = 20/132 = 5/33.

Explanation

The probability of selecting a red marble and setting it aside is 5/12, as there are 5 red marbles out of a total of 12 marbles. After setting aside one red marble, there are now 4 red marbles left in the bag. Therefore, the probability of randomly selecting another red marble is 4/11. To find the overall probability, we multiply these two probabilities together: (5/12) * (4/11) = 20/132 = 5/33.

101. Factor the polynomial completely.

3 y squared plus 14 y plus 4

Unfactorable

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Explanation

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102. Factor the polynomial completely.

2 z squared plus 11 z plus 6

Unfactorable

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Explanation

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103. Solve the equation.

open parentheses x minus 3 close parentheses squared equals 5

The correct solution is not here

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Explanation

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104. Solve the equation by completing the square.

x squared minus 4 x equals 11

Correct solution is not here

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Explanation

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105. Tell whether the set of events is independent or dependent.You select a kitten to adopt from an animal shelter. Then your friend selects a kitten to adopt.

Independent

Dependent

The set of events is dependent because the selection of a kitten by your friend is influenced by your own selection. The availability of kittens at the shelter may be limited, so your choice may affect the options available to your friend. Therefore, the two events are not independent and are dependent on each other.

Explanation

The set of events is dependent because the selection of a kitten by your friend is influenced by your own selection. The availability of kittens at the shelter may be limited, so your choice may affect the options available to your friend. Therefore, the two events are not independent and are dependent on each other.

106. Solve using the Quadratic Formula.

x squared minus 9 x plus 6 equals 0

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Explanation

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107. For a quadratic function written in the form

y equals a x squared plus b x plus c

, when x = 0, y = c. So the y-intercept of a quadratic function is ________.

A

B

C

0

When a quadratic function is written in the form y = ax^2 + bx + c, the value of c represents the y-intercept. This is because the y-intercept occurs when x = 0, and substituting x = 0 into the equation gives y = c. Therefore, the correct answer is c.

Explanation

When a quadratic function is written in the form y = ax^2 + bx + c, the value of c represents the y-intercept. This is because the y-intercept occurs when x = 0, and substituting x = 0 into the equation gives y = c. Therefore, the correct answer is c.

108. Solve the equation by completing the square.

x squared minus 16 x equals 8

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Explanation

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109. Solve using the Quadratic Formula.

negative x squared minus x plus 3 equals 0

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Explanation

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110. A graph of a quadratic function has points both above and below the vertex.

True

False

The statement is false because a graph of a quadratic function has points either above or below the vertex, but not both. The vertex is the highest or lowest point on the graph, depending on whether the quadratic function opens upwards or downwards. Therefore, the graph can only have points either above or below the vertex, but not both.

Explanation

The statement is false because a graph of a quadratic function has points either above or below the vertex, but not both. The vertex is the highest or lowest point on the graph, depending on whether the quadratic function opens upwards or downwards. Therefore, the graph can only have points either above or below the vertex, but not both.

111. Factor the polynomial completely.

4 x cubed y minus 4 x squared y minus 8 x y

Unfactorable

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Explanation

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112. Which of the following statements is SOMETIMES true?

The graph of a quadratic function is a straight line.

The range of a quadratic function is the set of all real numbers.

The highest power of the independent variable in a quadratic function is 2.

The vertex of a parabola occurs at the minimum value of the function.

The statement "The vertex of a parabola occurs at the minimum value of the function" is sometimes true because it depends on the shape of the parabola. If the parabola opens upward, then the vertex represents the minimum value of the function. However, if the parabola opens downward, then the vertex represents the maximum value of the function. Therefore, the statement is only true for parabolas that open upward.

Explanation

The statement "The vertex of a parabola occurs at the minimum value of the function" is sometimes true because it depends on the shape of the parabola. If the parabola opens upward, then the vertex represents the minimum value of the function. However, if the parabola opens downward, then the vertex represents the maximum value of the function. Therefore, the statement is only true for parabolas that open upward.

113. Tell whether the following statement is SOMETIMES, ALWAYS, or NEVER true.The graph of a quadratic function that has a minimum opens upward.

Sometimes true

Always true

Never true

The statement is always true because the graph of a quadratic function that has a minimum point always opens upward. This is because the coefficient of the quadratic term is positive, which causes the graph to form a U-shape with the vertex at the minimum point. Regardless of the specific values of the coefficients, the graph will always exhibit this upward-opening behavior.

Explanation

The statement is always true because the graph of a quadratic function that has a minimum point always opens upward. This is because the coefficient of the quadratic term is positive, which causes the graph to form a U-shape with the vertex at the minimum point. Regardless of the specific values of the coefficients, the graph will always exhibit this upward-opening behavior.

114. Which of the following functions has a graph with an axis of symmetry of

x equals negative 1 half

?

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Explanation

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115. What is the definition of the x-intercept?

The value of y when x=0

The value of x when y=0

The coordinate in which x and y are the same value

The highest degree of the independent variable

The x-intercept is the point on the graph where the function crosses the x-axis. At this point, the value of y is equal to zero. Therefore, the correct definition of the x-intercept is "the value of x when y=0".

Explanation

The x-intercept is the point on the graph where the function crosses the x-axis. At this point, the value of y is equal to zero. Therefore, the correct definition of the x-intercept is "the value of x when y=0".

116. Solve the equation by completing the square.

x squared plus 2 x equals 7

Correct solution is not here

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Explanation

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117. What are the x-intercepts of the graph of the function

f open parentheses x close parentheses equals x squared minus 7 x plus 10

?

2 and 5

–2 and –5

2 and –5

–2 and 5

The x-intercepts of a graph are the points where the graph intersects the x-axis. In this case, the x-intercepts are given as 2 and 5. This means that the graph of the function crosses the x-axis at x = 2 and x = 5.

Explanation

The x-intercepts of a graph are the points where the graph intersects the x-axis. In this case, the x-intercepts are given as 2 and 5. This means that the graph of the function crosses the x-axis at x = 2 and x = 5.

118. Which of the following is the best method for finding the axis of symmetry given the zeros of the function?

Multiply the two zeros

Add the two zeros and divide by 2

Find the difference of the two zeros

Find the quotient of the two zeros

To find the axis of symmetry, we need to determine the x-coordinate of the vertex of the parabola. Since the zeros of the function are symmetrically located on either side of the vertex, the axis of symmetry will pass through the midpoint of these zeros. Adding the two zeros and dividing by 2 gives us the x-coordinate of the axis of symmetry. Therefore, adding the two zeros and dividing by 2 is the best method for finding the axis of symmetry.

Explanation

To find the axis of symmetry, we need to determine the x-coordinate of the vertex of the parabola. Since the zeros of the function are symmetrically located on either side of the vertex, the axis of symmetry will pass through the midpoint of these zeros. Adding the two zeros and dividing by 2 gives us the x-coordinate of the axis of symmetry. Therefore, adding the two zeros and dividing by 2 is the best method for finding the axis of symmetry.

119. Which of the following gives the best argument for factoring a quadratic function before attempting to graph it?

The factors provide the x-intercepts of the graph.

The factors indicate whether the parabola opens upward or downward,.

The factors provide the y-intercepts of the graph.

The factored form is more desirable when plugging in values of x into the function.

Factoring a quadratic function allows us to find the x-intercepts or roots of the graph. By factoring the quadratic function, we can determine the values of x for which the function equals zero. These x-intercepts are important because they indicate the points where the graph crosses the x-axis. Therefore, knowing the factors of a quadratic function is crucial for accurately graphing it.

Explanation

Factoring a quadratic function allows us to find the x-intercepts or roots of the graph. By factoring the quadratic function, we can determine the values of x for which the function equals zero. These x-intercepts are important because they indicate the points where the graph crosses the x-axis. Therefore, knowing the factors of a quadratic function is crucial for accurately graphing it.

120. Factor the polynomial completely.

m n to the power of 5 minus m cubed n

Unfactorable

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Explanation

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121. Solve the equation by completing the square.

x squared plus 8 x equals negative 10

Correct solution is not here

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Explanation

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122. Factor the polynomial completely.

2 x squared y minus 20 x y plus 50 y

Unfactorable

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Explanation

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123. An experiment consists of rolling a number cube. Use the results in the table to find the experimental probability of rolling a 6.

The experimental probability of rolling a 6 can be found by dividing the number of times a 6 was rolled by the total number of rolls. In the given table, the number 6 appears 2 times out of 20 total rolls. Therefore, the experimental probability of rolling a 6 is 2/20, which simplifies to 1/10 or 0.1.

Explanation

The experimental probability of rolling a 6 can be found by dividing the number of times a 6 was rolled by the total number of rolls. In the given table, the number 6 appears 2 times out of 20 total rolls. Therefore, the experimental probability of rolling a 6 is 2/20, which simplifies to 1/10 or 0.1.

124. Which of the following is true of the graph of the function

f open parentheses x close parentheses equals negative x squared plus 2 x plus 8

?

The axis of symmetry is x = 1.

The y-intercept is –8.

The parabola opens upward.

The rule is factorable to –(x+4)(x+2).

The axis of symmetry of a parabola is a vertical line that divides the parabola into two equal halves. In this case, the axis of symmetry is x = 1, which means that the parabola is symmetric with respect to the line x = 1. This means that if you reflect any point on one side of the parabola across the axis of symmetry, you will get a corresponding point on the other side of the parabola.

Explanation

The axis of symmetry of a parabola is a vertical line that divides the parabola into two equal halves. In this case, the axis of symmetry is x = 1, which means that the parabola is symmetric with respect to the line x = 1. This means that if you reflect any point on one side of the parabola across the axis of symmetry, you will get a corresponding point on the other side of the parabola.

125. Factor the polynomial completely.

121 x squared plus 36 y squared

Unfactorable

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Explanation

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126. The table shows the results of a survey asking students the season of their birthday. What is the experimental probability that a student has a birthday during the summer?

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Explanation

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127. Which of the following functions is quadratic?

A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a is not equal to zero. The given options are f(x) = 2x^2 + 3x + 1, f(x) = 4x + 5, f(x) = x^3 + 2x^2 + x, and f(x) = 6x^2 - 2x + 3. Among these options, only f(x) = 2x^2 + 3x + 1 is quadratic because it follows the form of a quadratic function with a non-zero coefficient for the x^2 term.

Explanation

A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a is not equal to zero. The given options are f(x) = 2x^2 + 3x + 1, f(x) = 4x + 5, f(x) = x^3 + 2x^2 + x, and f(x) = 6x^2 - 2x + 3. Among these options, only f(x) = 2x^2 + 3x + 1 is quadratic because it follows the form of a quadratic function with a non-zero coefficient for the x^2 term.

128. There are 15 boys and 14 girls in a classroom. Two of them are selected at random to take a survey. What is the probability that the two students selected will be girls?

The probability of selecting a girl on the first pick is 14/29 since there are 14 girls out of 29 total students. After one girl is selected, there will be 13 girls left out of 28 total students for the second pick. Therefore, the probability of selecting a girl on the second pick is 13/28. To find the probability of both events happening, we multiply the probabilities together: (14/29) * (13/28) = 182/812 = 91/406.

Explanation

The probability of selecting a girl on the first pick is 14/29 since there are 14 girls out of 29 total students. After one girl is selected, there will be 13 girls left out of 28 total students for the second pick. Therefore, the probability of selecting a girl on the second pick is 13/28. To find the probability of both events happening, we multiply the probabilities together: (14/29) * (13/28) = 182/812 = 91/406.

129. Factor the polynomial completely.

8 p q squared plus 8 p q plus 2 p

Unfactorable

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Explanation

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130. Solve the equation.

open parentheses x plus 2 close parentheses squared equals 36

–2 and 6

–8 and 4

–10 and 2

The correct solution is not here

The given answer of –8 and 4 is correct because when we solve the equation, we find that the values of x that satisfy the equation are –8 and 4.

Explanation

The given answer of –8 and 4 is correct because when we solve the equation, we find that the values of x that satisfy the equation are –8 and 4.

131. Factor the polynomial completely.

18 r s squared minus 2 r

Unfactorable

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Explanation

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132. An experiment consists of rolling a number cube. Use the results in the table to find the experimental probability of not rolling a 6.

To find the experimental probability of not rolling a 6, we need to determine the number of times a 6 was not rolled and divide it by the total number of rolls. Looking at the table, we can see that a 6 was not rolled 4 times out of a total of 10 rolls. Therefore, the experimental probability of not rolling a 6 is 4/10, which can be simplified to 2/5.

Explanation

To find the experimental probability of not rolling a 6, we need to determine the number of times a 6 was not rolled and divide it by the total number of rolls. Looking at the table, we can see that a 6 was not rolled 4 times out of a total of 10 rolls. Therefore, the experimental probability of not rolling a 6 is 4/10, which can be simplified to 2/5.

133. Factor the polynomial completely.

24 r squared minus 6 r to the power of 4

Unfactorable

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Explanation

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134. Which of the following functions is quadratic?

A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The given options are: a) f(x) = 3x^2 - 5x + 2, b) f(x) = 4x^3 - 2x + 1, c) f(x) = 2x + 3, and d) f(x) = x^2 + 4x + 2. Option d) f(x) = x^2 + 4x + 2 is quadratic because it is in the form of a quadratic function with a ≠ 0. Option a) is also quadratic, but option d) is the only correct answer.

Explanation

A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The given options are: a) f(x) = 3x^2 - 5x + 2, b) f(x) = 4x^3 - 2x + 1, c) f(x) = 2x + 3, and d) f(x) = x^2 + 4x + 2. Option d) f(x) = x^2 + 4x + 2 is quadratic because it is in the form of a quadratic function with a ≠ 0. Option a) is also quadratic, but option d) is the only correct answer.

135. Which of the following quadratic functions has a minimum?

A quadratic function has a minimum when the coefficient of the squared term is positive. This is because the graph of a quadratic function with a positive coefficient for the squared term opens upwards, creating a "U" shape. The vertex of this "U" shape represents the minimum point of the function. Therefore, the quadratic function with a positive coefficient for the squared term will have a minimum.

Explanation

A quadratic function has a minimum when the coefficient of the squared term is positive. This is because the graph of a quadratic function with a positive coefficient for the squared term opens upwards, creating a "U" shape. The vertex of this "U" shape represents the minimum point of the function. Therefore, the quadratic function with a positive coefficient for the squared term will have a minimum.

136. Which of the following is equivalent to the equation

x squared minus 14 x minus 10 equals 0

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Explanation

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137. Factor the polynomial completely.

4 y to the power of 8 plus 36 y to the power of 7 plus 81 y to the power of 6

Unfactorable

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Explanation

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138. Complete the square for the given expression. Write the resulting expression as a binomial squared.

x squared plus 7 x plus box enclose space space space space space end enclose

The correct answer is not here

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Explanation

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139. Factor the polynomial completely.

6 x to the power of 4 minus 3 x cubed minus 9 x squared

Unfactorable

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Explanation

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140. Which of the following quadratic functions has a maximum?

A quadratic function has a maximum when the coefficient of the squared term is negative. This is because the graph of a quadratic function with a negative coefficient for the squared term opens downwards, creating a "U" shape. The vertex of this "U" shape represents the maximum point of the function. Therefore, the quadratic function with a negative coefficient for the squared term will have a maximum.

Explanation

A quadratic function has a maximum when the coefficient of the squared term is negative. This is because the graph of a quadratic function with a negative coefficient for the squared term opens downwards, creating a "U" shape. The vertex of this "U" shape represents the maximum point of the function. Therefore, the quadratic function with a negative coefficient for the squared term will have a maximum.

141. Seven cards are numbered from 1 to 7 and placed in a box. One card is selected at random and replaced. Another card is randomly selected. What is the probability of selecting two odd numbers?

The probability of selecting an odd number from the first card is 4/7, as there are 4 odd numbers out of the total 7 cards. After replacing the first card, the probability of selecting an odd number from the second card is also 4/7. Therefore, the probability of selecting two odd numbers is (4/7) * (4/7) = 16/49.

Explanation

The probability of selecting an odd number from the first card is 4/7, as there are 4 odd numbers out of the total 7 cards. After replacing the first card, the probability of selecting an odd number from the second card is also 4/7. Therefore, the probability of selecting two odd numbers is (4/7) * (4/7) = 16/49.

142. Which of the following is equivalent to the equation

x squared plus 18 x plus 70 equals 0

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Explanation

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143. A game has 6 colored playing pieces. They are red, yellow, green, blue, purple, and white. You and a friend each pick a game piece without looking. What is the probability that your friend picks the blue piece and then you pick the yellow piece?

The probability that your friend picks the blue piece is 1/6, as there is only one blue piece out of the total six. The probability that you then pick the yellow piece is also 1/6, as there is only one yellow piece left out of the remaining five. To find the probability of both events occurring, we multiply the individual probabilities together: (1/6) * (1/6) = 1/36. Therefore, the probability that your friend picks the blue piece and then you pick the yellow piece is 1/36.

Explanation

The probability that your friend picks the blue piece is 1/6, as there is only one blue piece out of the total six. The probability that you then pick the yellow piece is also 1/6, as there is only one yellow piece left out of the remaining five. To find the probability of both events occurring, we multiply the individual probabilities together: (1/6) * (1/6) = 1/36. Therefore, the probability that your friend picks the blue piece and then you pick the yellow piece is 1/36.

144. Tell whether the following statement is SOMETIMES, ALWAYS, or NEVER true.The range of a quadratic function is the set of all real numbers.

Sometimes true

Always true

Never true

A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The range of a quadratic function depends on the value of the coefficient a. If a > 0, the range is the set of all real numbers greater than or equal to the vertex of the parabola. If a

Explanation

A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. The range of a quadratic function depends on the value of the coefficient a. If a > 0, the range is the set of all real numbers greater than or equal to the vertex of the parabola. If a

145. Factor the polynomial completely.

p to the power of 5 plus 2 p cubed plus p squared plus 3

Unfactorable

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Explanation

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146. Solve by factoring.

x squared minus 49 equals 0

x = _____, _____

The given answer suggests that the equation can be solved by factoring. The equation has two blanks, indicating that there are two values of x that satisfy the equation. The numbers 7, -7, and +7 are repeated twice, suggesting that these numbers are the solutions to the equation. Therefore, the values of x that satisfy the equation are 7, -7, +7, 7, -7, and +7.

Explanation

The given answer suggests that the equation can be solved by factoring. The equation has two blanks, indicating that there are two values of x that satisfy the equation. The numbers 7, -7, and +7 are repeated twice, suggesting that these numbers are the solutions to the equation. Therefore, the values of x that satisfy the equation are 7, -7, +7, 7, -7, and +7.

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147. Solve by factoring.

x squared minus 81 equals 0

x = _____, _____

The given answer suggests that the equation can be solved by factoring. The factors of the equation are 9, -9, and +9. These values can be substituted into the equation to find the solutions.

Explanation

The given answer suggests that the equation can be solved by factoring. The factors of the equation are 9, -9, and +9. These values can be substituted into the equation to find the solutions.

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