Question

which equation could be used to find the zeros of the function f(x) = 2x² + 5x - 12? (a) (x + 6)(2x - 2) = 0 (b) (2x + 3)(x - 4) = 0 (c) (2x - 3)(x + 4) = 0 (d) (2x + 6)(x - 2) = 0

Explanation:

Step1: Recall factoring quadratic

To find the zeros, factor \(f(x) = 2x^2 + 5x - 12\). We need two numbers \(a,b\) such that \(a\times b=2\times(-12)= - 24\) and \(a + b=5\). The numbers are \(8\) and \(- 3\) (since \(8\times(-3)=-24\) and \(8+( - 3)=5\)).

Step2: Rewrite middle term

Rewrite \(5x\) as \(8x-3x\): \(2x^2+8x - 3x - 12\).

Step3: Factor by grouping

Group terms: \((2x^2 + 8x)+(-3x - 12)\). Factor out common factors: \(2x(x + 4)-3(x + 4)\). Then factor out \((x + 4)\): \((2x - 3)(x + 4)\). Set equal to zero: \((2x - 3)(x + 4)=0\).

Answer:

(c) \((2x - 3)(x + 4)=0\)