TExES Core Mathematics Linear and Quadratic Equations

1. Solve for x: 3x + 7 = 22

X = 5

X = 9

X = 3

X = 7

To solve the equation 3x + 7 = 22, first subtract 7 from both sides to get 3x = 15. Then, divide both sides by 3 to isolate x, resulting in x = 5. This shows that when x is 5, the original equation holds true.

Explanation

To solve the equation 3x + 7 = 22, first subtract 7 from both sides to get 3x = 15. Then, divide both sides by 3 to isolate x, resulting in x = 5. This shows that when x is 5, the original equation holds true.

2. Which equation represents a line with slope 2 and y-intercept –3?

Y = 2x + 3

Y = 2x – 3

Y = –2x – 3

Y = 3x – 2

A line with a slope of 2 means that for every unit increase in x, y increases by 2. The y-intercept of -3 indicates that the line crosses the y-axis at (0, -3). Therefore, the equation y = 2x - 3 correctly represents these characteristics, showing the relationship between x and y.

Explanation

A line with a slope of 2 means that for every unit increase in x, y increases by 2. The y-intercept of -3 indicates that the line crosses the y-axis at (0, -3). Therefore, the equation y = 2x - 3 correctly represents these characteristics, showing the relationship between x and y.

3. Solve the system: x + y = 5 and 2x – y = 4

X = 3, y = 2

X = 2, y = 3

X = 1, y = 4

X = 4, y = 1

To solve the system of equations, substitute one equation into the other. From x + y = 5, we can express y as 5 - x. Substituting this into the second equation (2x - y = 4) leads to 2x - (5 - x) = 4. Simplifying gives x = 3, and substituting back reveals y = 2.

Explanation

To solve the system of equations, substitute one equation into the other. From x + y = 5, we can express y as 5 - x. Substituting this into the second equation (2x - y = 4) leads to 2x - (5 - x) = 4. Simplifying gives x = 3, and substituting back reveals y = 2.

4. Factor: x² + 7x + 12

(x + 3)(x + 4)

(x + 2)(x + 6)

(x + 1)(x + 12)

(x + 5)(x + 2)

To factor the quadratic expression x² + 7x + 12, we need two numbers that multiply to 12 (the constant term) and add up to 7 (the coefficient of x). The numbers 3 and 4 satisfy these conditions, leading to the factors (x + 3) and (x + 4).

Explanation

To factor the quadratic expression x² + 7x + 12, we need two numbers that multiply to 12 (the constant term) and add up to 7 (the coefficient of x). The numbers 3 and 4 satisfy these conditions, leading to the factors (x + 3) and (x + 4).

5. Solve x² – 5x + 6 = 0

X = 2 or x = 3

X = 1 or x = 6

X = –2 or x = –3

X = 0 or x = 5

To solve the quadratic equation x² – 5x + 6 = 0, we can factor it as (x - 2)(x - 3) = 0. Setting each factor to zero gives the solutions x = 2 and x = 3. These values satisfy the original equation, confirming them as the correct answers.

Explanation

To solve the quadratic equation x² – 5x + 6 = 0, we can factor it as (x - 2)(x - 3) = 0. Setting each factor to zero gives the solutions x = 2 and x = 3. These values satisfy the original equation, confirming them as the correct answers.

6. What is the vertex of the parabola y = (x – 2)² + 3?

(2, 3)

(–2, 3)

(2, –3)

(–2, –3)

The vertex of a parabola in the form \(y = (x - h)^2 + k\) is given by the point \((h, k)\). In this equation, \(h = 2\) and \(k = 3\), thus the vertex is at the point \((2, 3)\). This point represents the minimum value of the parabola, as it opens upwards.

Explanation

The vertex of a parabola in the form \(y = (x - h)^2 + k\) is given by the point \((h, k)\). In this equation, \(h = 2\) and \(k = 3\), thus the vertex is at the point \((2, 3)\). This point represents the minimum value of the parabola, as it opens upwards.

7. Use the quadratic formula to solve 2x² – 3x – 2 = 0. Which is a solution?

X = 2

X = –1/2

X = 3

X = 1

To solve the quadratic equation 2x² – 3x – 2 = 0 using the quadratic formula, we identify coefficients a = 2, b = -3, and c = -2. Plugging these values into the formula, we calculate the roots and find that x = 2 is one of the solutions.

Explanation

To solve the quadratic equation 2x² – 3x – 2 = 0 using the quadratic formula, we identify coefficients a = 2, b = -3, and c = -2. Plugging these values into the formula, we calculate the roots and find that x = 2 is one of the solutions.

8. Solve: |2x – 5| = 7

X = 6 or x = –1

X = 1 or x = 6

X = –1 or x = 5

X = 2 or x = 5

To solve the equation |2x – 5| = 7, we consider two cases: 2x – 5 = 7 and 2x – 5 = -7. Solving these gives us 2x = 12 (x = 6) and 2x = -2 (x = -1). Thus, the solutions are x = 6 and x = -1.

Explanation

To solve the equation |2x – 5| = 7, we consider two cases: 2x – 5 = 7 and 2x – 5 = -7. Solving these gives us 2x = 12 (x = 6) and 2x = -2 (x = -1). Thus, the solutions are x = 6 and x = -1.

9. Find the x-intercepts of y = x² – 4

X = 2 and x = –2

X = 4 and x = –4

X = 0 and x = 4

X = 1 and x = –1

To find the x-intercepts of the equation y = x² – 4, set y to zero: 0 = x² – 4. This simplifies to x² = 4, leading to x = ±2. Thus, the x-intercepts are at the points (2, 0) and (-2, 0).

Explanation

To find the x-intercepts of the equation y = x² – 4, set y to zero: 0 = x² – 4. This simplifies to x² = 4, leading to x = ±2. Thus, the x-intercepts are at the points (2, 0) and (-2, 0).

10. Simplify: (2x – 3)(x + 4)

2x² + 5x – 12

2x² + 11x – 12

2x² + 8x – 12

2x² – 5x – 12

To simplify the expression (2x – 3)(x + 4), apply the distributive property (FOIL method). Multiply the first terms (2x * x), the outer terms (2x * 4), the inner terms (-3 * x), and the last terms (-3 * 4). Combine like terms to arrive at 2x² + 8x - 3x - 12, which simplifies to 2x² + 11x - 12.

Explanation

To simplify the expression (2x – 3)(x + 4), apply the distributive property (FOIL method). Multiply the first terms (2x * x), the outer terms (2x * 4), the inner terms (-3 * x), and the last terms (-3 * 4). Combine like terms to arrive at 2x² + 8x - 3x - 12, which simplifies to 2x² + 11x - 12.

11. Solve: 4(x – 2) = 2(x + 3)

X = 7

X = 5

X = 9

X = 3

To solve the equation 4(x – 2) = 2(x + 3), first distribute the constants: 4x - 8 = 2x + 6. Next, isolate x by moving terms involving x to one side and constants to the other: 4x - 2x = 6 + 8, leading to 2x = 14. Finally, divide by 2 to find x = 7.

Explanation

To solve the equation 4(x – 2) = 2(x + 3), first distribute the constants: 4x - 8 = 2x + 6. Next, isolate x by moving terms involving x to one side and constants to the other: 4x - 2x = 6 + 8, leading to 2x = 14. Finally, divide by 2 to find x = 7.

12. What is the discriminant of 3x² – 6x + 2 = 0?

12

24

36

0

The discriminant of a quadratic equation in the form ax² + bx + c is calculated using the formula D = b² - 4ac. For the equation 3x² – 6x + 2, a = 3, b = -6, and c = 2. Substituting these values gives D = (-6)² - 4(3)(2) = 36 - 24 = 12.

Explanation

The discriminant of a quadratic equation in the form ax² + bx + c is calculated using the formula D = b² - 4ac. For the equation 3x² – 6x + 2, a = 3, b = -6, and c = 2. Substituting these values gives D = (-6)² - 4(3)(2) = 36 - 24 = 12.