Algebra 2 Final Exam Review Test

Explanation
The given expression x2 - 9 can be factored using the difference of squares formula. This formula states that a2 - b2 can be factored as (a - b)(a + b). In this case, a is x and b is 3. Therefore, x2 - 9 can be factored as (x - 3)(x + 3). However, the given answer is (x-3)(x-3), which is incorrect.

Explanation
The given expression is a difference of squares, which can be factored using the formula a^2 - b^2 = (a+b)(a-b). In this case, a = 4x and b = 5. Plugging in these values, we get (4x)^2 - 5^2 = (4x+5)(4x-5). Therefore, the correct answer is (4x-5)(4x+5).

Explanation
The given expression can be factored into the form (x+3)(3x+7). This can be determined by using the distributive property to expand the product of the two binomials, which results in 3x^2 + 7x + 9x + 21. Combining like terms, we get 3x^2 + 16x + 21, which matches the given expression. Therefore, the correct answer is (x+3)(3x+7).

Explanation
The given equation x^2 - 9x + 18 = 0 can be factored as (x-3)(x-6) = 0. By setting each factor equal to zero, we find that x = 3 and x = 6 are the solutions to the equation.

Explanation
The given equation is a quadratic equation. To solve it, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a). In this case, a = 8, b = 8, and c = -6. Plugging these values into the quadratic formula, we get x = (-8 ± √(8^2 - 4(8)(-6))) / (2(8)). Simplifying further, we have x = (-8 ± √(64 + 192)) / 16. Continuing to simplify, we get x = (-8 ± √256) / 16. Finally, we have x = (-8 ± 16) / 16, which gives us two solutions: x = -3/2 and x = 1/2.

Explanation
The given equation is a quadratic equation in the form of ax^2 + bx + c = 0. In this case, the equation is 5x^2 - 40 = 0. To solve this equation, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. By substituting the values of a, b, and c from the given equation into the quadratic formula, we get x = (+/- √(40^2 - 4(5)(-40))) / (2(5)). Simplifying further, we have x = (+/- √(1600 + 800)) / 10, which gives us x = (+/- √2400) / 10. Taking the square root of 2400, we get x = (+/- 48) / 10, which simplifies to x = +/- 8. Therefore, the correct answer is x = +/- 8.

Explanation
The given equation can be rewritten as -9 + 6 = 3i + 6. Simplifying further, we get -3 = 3i. To express this in the form a + bi, we can write it as 0 + (-3)i. Therefore, the answer is 6 + 3i, a + bi, where a = 6 and b = 3.

Explanation
The given expression is a combination of square roots and complex numbers. It simplifies to 7 + 3i√2.

Explanation
The given expression is a sum of two complex numbers. To simplify, we add the real parts and the imaginary parts separately. The real parts are 6 and 5, so their sum is 11. The imaginary parts are -3i and 4i, so their sum is 1i. Therefore, the simplified form of the expression is 11 + 1i.

Explanation
The given expression is a subtraction of two complex numbers. To solve it, we subtract the real parts and the imaginary parts separately. (-1 + i) - (7 - 5i) = -1 - 7 + (1 + 5)i = -8 + 6i. Therefore, the correct answer is -8 + 6i.

Explanation
The given expression involves multiplying 3i with (-8-2i). To solve this, we use the distributive property of multiplication over addition. First, we multiply 3i with -8, which gives us -24i. Then, we multiply 3i with -2i, which gives us -6i^2. Since i^2 is equal to -1, we can substitute -6i^2 with 6. Therefore, the final result is -24i + 6, which can be simplified as 6 - 24i.

Explanation
The given expression is a multiplication of two complex numbers. To multiply complex numbers, we use the distributive property and combine like terms. In this case, we have (-2+5i) multiplied by (-1+4i). When we multiply these two complex numbers, we get -18-13i as the result. Hence, the answer -18-13i is the correct result of the given multiplication.

Explanation
The correct answer is x= -1,3 because when we substitute -1 and 3 into the equation x2 + 4x = -3, we get (-1)2 + 4(-1) = 1 - 4 = -3 and (3)2 + 4(3) = 9 + 12 = 21, which satisfies the equation. Therefore, -1 and 3 are the values of x that make the equation true.

Explanation
The equation x^2 - 12x = -27 can be rewritten as x^2 - 12x + 27 = 0. To solve this quadratic equation, we can factor it as (x-9)(x-3) = 0. This means that either (x-9) = 0 or (x-3) = 0. Solving these equations, we find that x = 9 and x = 3 are the solutions to the equation x^2 - 12x = -27. Therefore, the answer is x = 9, 3.

Explanation
The equation x^2 + 8x = 0 can be factored as x(x + 8) = 0. This equation is satisfied when either x = 0 or x + 8 = 0. Therefore, the solutions to the equation are x = 0 and x = -8.

Explanation
The given equation x^2 + 14x + 48 = 5 can be solved by subtracting 5 from both sides to get x^2 + 14x + 43 = 0. This is a quadratic equation that can be factored as (x + 7)(x + 6) = 0. Setting each factor equal to zero gives x = -7 and x = -6. Therefore, the solution to the equation is x = -7 +/- sqrt(6).

Explanation
The quadratic equation given is x^2 - 10x = 0. To solve this equation, we can use the quadratic formula. The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions for x can be found using the formula x = (-b ± √(b^2 - 4ac)) / (2a). In this case, a = 1, b = -10, and c = 0. Plugging these values into the quadratic formula, we get x = (10 ± √(10^2 - 4(1)(0))) / (2(1)). Simplifying this expression gives us x = (10 ± √(100)) / 2. Therefore, x = (10 ± 10) / 2, which simplifies to x = 0 and x = 10.

Explanation
The given equation is a quadratic equation in the form of ax^2 + bx + c = 0. To find the solutions, we can use the quadratic formula, which states that x = (-b ± √(b^2 - 4ac)) / (2a). In this case, the equation is x^2 + 12x + 20 = 0, so a = 1, b = 12, and c = 20. Plugging these values into the quadratic formula, we get x = (-12 ± √(12^2 - 4(1)(20))) / (2(1)). Simplifying further, we have x = (-12 ± √(144 - 80)) / 2, which becomes x = (-12 ± √64) / 2. Finally, x = (-12 ± 8) / 2, giving us the solutions x = -2 and x = -10.

Explanation
The given expression is in standard polynomial form. It consists of terms with different powers of x. The term -x^3 represents a cubic term with a coefficient of -1, and the constant term is 2. Therefore, the correct answer is -x^3 + 2.

Explanation
The given expression (x-5)^3 can be expanded using the binomial theorem. This states that (a+b)^n = a^n + n*a^(n-1)*b + (n(n-1)/2)*a^(n-2)*b^2 + ... + b^n. In this case, a = x and b = -5. Plugging these values into the binomial theorem, we get x^3 + 3*x^2*(-5) + 3*x*(-5)^2 + (-5)^3. Simplifying this expression gives x^3 - 15x^2 + 75x - 125, which matches the given answer.

Explanation
The given equation is a quadratic equation. By solving it, we find that the values of x that satisfy the equation are x=7 and x= +/- sqrt(2). These values are obtained by factoring the equation or by using the quadratic formula. Therefore, the correct answer is x=7, x= +/- sqrt(2).

Explanation
The given equation is 27x3 - 1 = 0. To solve for x, we need to isolate x on one side of the equation. Adding 1 to both sides, we get 27x3 = 1. Then, dividing both sides by 27, we find x3 = 1/27. Taking the cube root of both sides, we obtain x = 1/3. Therefore, the correct answer is x = 1/3.

Explanation
The given expression x3 + 8 can be factored as (x + 2)(x2 - 2x + 4), which is the answer provided. This can be determined by recognizing that the given expression is a sum of cubes, where 8 can be written as 2^3. Using the formula for the sum of cubes, we can factor it as (x + 2)(x2 - 2x + 4). Therefore, the correct answer is (x + 2)(x2 - 2x + 4).

Explanation
The given equation x3 - 64 = 0 can be rewritten as (x - 4)(x2 + 4x + 16) = 0. This equation can be solved by setting each factor equal to zero. Therefore, x - 4 = 0, which gives x = 4 as the solution.

Explanation
The given polynomial long division is dividing x^2 + 3x - 12 by x - 3. The quotient obtained is x + 6 and the remainder is 6. This means that when we divide x^2 + 3x - 12 by x - 3, the result is x + 6 with a remainder of 6.

Explanation
The given question asks to perform synthetic division on the polynomial 3x^3 - 4x^2 + 2x - 1 divided by x + 1. Synthetic division is a method used to divide polynomials. The answer 3x^2 - 7x + 9, R -10 represents the quotient and remainder obtained after performing synthetic division. The quotient is 3x^2 - 7x + 9 and the remainder is -10.