What are the roots of x2 + 8x – 48 = 0?
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Solution
The roots of a quadratic are the solutions of the quadratic. Factor the quadratic and set each factor equal to 0 to find the roots. To find the factors of a quadratic, begin by finding two numbers whose product is equal to the constant of the quadratic. Of those numbers, find the pair that adds to the coefficient of the x term of the quadratic; –4 and 12 multiply to –48 and add to 8. Therefore, the factors of x2 + 8x – 48 are (x – 4) and (x + 12). Set each factor equal to 0 and solve for x: x – 4 = 0, x = 4, and x + 12 = 0, x = –12. The roots of x2 + 8x – 48 are 4 and –12.
What are the roots of x2 – 18x + 32 = 0?
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Solution
The roots of a quadratic are the solutions of the quadratic. Factor the quadratic and set each factor equal to 0 to find the roots. To find the factors of a quadratic, begin by finding two numbers whose product is equal to the constant of the quadratic. Of those numbers, find the pair that adds to the coefficient of the x term of the quadratic; –2 and –16 multiply to 32 and add to –18. Therefore, the factors of x2 – 18x + 32 are (x – 2) and (x – 16). Set each factor equal to 0 and solve for x: x – 2 = 0, x = 2, and x – 16 = 0, x = 16. The roots of x2 – 18x + 32 are 2 and 16.
What are the factors of x2 – 11x + 28?
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Solution
To find the factors of a quadratic, begin by finding two numbers whose product is equal to the constant of the quadratic. Of those numbers, find the pair that adds to the coefficient of the x term of the quadratic; –4 and –7 multiply to 28 and add to –11. Therefore, the factors of x2 – 11x + 28 are (x – 4) and (x – 7).
What is one factor of x2 – 4?
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Solution
To find the factors of a quadratic, begin by finding two numbers whose product is equal to the constant of the quadratic. Of those numbers, find the pair that adds to the coefficient of the x term of the quadratic. This quadratic has no x term—the sum of the products of the outside and inside terms of the factors is 0; –2 and 2 multiply to –4 and add to 0. Therefore, the factors of x2 – 4 are (x – 2) and (x + 2).
What are the factors of x2 – x – 6?
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Solution
To find the factors of a quadratic, begin by finding two numbers whose product is equal to the constant of the quadratic. Of those numbers, find the pair that adds to the coefficient of the x term of the quadratic; –3 and 2 multiply to –6 and add to –1. Therefore, the factors of x2 – x – 6 are (x – 3) and (x + 2).
What are the factors of 2x3 + 8x2 – 192x?
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Solution
To find the product of two binomials, multiply the first term of each binomial, the outside terms, the inside terms, and the last terms. Then, add the products; (2x + 6)(3x – 9) = 6x2 – 18x + 18x – 54 = 6x2 – 54
What are the factors of 64x3 – 16x?
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Solution
(x + c)2 = (x + c)(x + c). To find the product of two binomials, multiply the first term of each binomial, the outside terms, the inside terms, and the last terms. Then, add the products; (x + c)(x + c) = x2 + cx + cx + c2 = x2 + 2cx + c2.
(x – 6)(x – 3)(x – 1) =
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Solution
To find the product of two binomials,multiply the first term of each binomial, the outside terms, the inside terms, and the last terms. Then, add the products; (x – 1)(x + 1) = x2 + x – x – 1 = x2 – 1.
(x2 + 5x – 7)(x + 2) =
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Solution
To find the product of two binomials,multiply the first term of each binomial, the outside terms, the inside terms, and the last terms. Then, add the products; (x – 6)(x – 6) = x2 – 6x – 6x + 36 = x2 – 12x + 36.
–3x(x + 6)(x – 9) =
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Solution
To find the product of two binomials, multiply the first term of each binomial, the outside terms, the inside terms, and the last terms. Then, add the products; (x – 3)(x + 7) = x2 + 7x – 3x – 21 = x2 + 4x – 21.