Factor the following quadratic expressions, if possible. (a). k^(2)-12 k+20 (b). 6x^(2)+17 x-14 (c). x^(2)-8x+16 (d). quad9m^(2)-1 (e). Parts (a) through (e) are trinomials while part (d) is a binomial, yet they are all quadratics. What makes each of them a quadratic?

Question

Bytelearn · Accepted Answer

Factor Quadratic Expression: a. Factor the quadratic expression k^2 - 12k + 20. To factor a quadratic expression of the form ax^2 + bx + c, we need to find two numbers that multiply to ac (the product of the coefficient of x^2 and the constant term) and add up to b (the coefficient of x). For k^2 - 12k + 20, we have a = 1, b = -12, and c = 20. We need two numbers that multiply to 20 and add up to -12. The numbers -10 and -2 satisfy these conditions. So, we can write the expression as (k - 10)(k - 2).Factor Quadratic Expression: b. Factor the quadratic expression 6x^2 + 17x - 14. For 6x^2 + 17x - 14, we have a = 6, b = 17, and c = -14. We need two numbers that multiply to -84 (6 * -14) and add up to 17. The numbers 21 and -4 satisfy these conditions. So, we can write the expression as (6x - 4)(x + 21/6). However, we need to simplify the second factor to get integer coefficients. Dividing 21 by 6 gives 3.5, so the factorization is (3x - 2)(2x + 7).Factor Quadratic Expression: c. Factor the quadratic expression x^2 - 8x + 16. For x^2 - 8x + 16, we have a = 1, b = -8, and c = 16. We need two numbers that multiply to 16 and add up to -8. The numbers -4 and -4 satisfy these conditions. So, we can write the expression as (x - 4)^2.Factor Quadratic Expression: d. Factor the quadratic expression 9m^2 - 1. The expression 9m^2 - 1 is a difference of squares, which can be factored as (a^2 - b^2) = (a + b)(a - b). Here, a = 3m and b = 1. So, we can write the expression as (3m + 1)(3m - 1).Explanation of Quadratics: e. Explain why parts (a) through (e) are all quadratics. A quadratic expression is an algebraic expression of degree 2, which means the highest power of the variable is 2. In parts (a) through (c), the expressions are trinomials with the highest power of the variable being 2. In part (d), the expression is a binomial with the highest power of the variable being 2. Therefore, all parts are quadratic expressions.

Full solution

More problems from Find derivatives of using multiple formulae

Related topics