Question

use grouping to factor and solve the quadratic equation: $8x^2 + 6x - 9 = 0$.

a. $x = \frac{3}{2}, x = -\frac{2}{3}$

b. $x = -\frac{3}{4}, x = \frac{3}{2}$

c. $x = -\frac{3}{2}, x = \frac{2}{3}$

d. $x = \frac{3}{4}, x = -\frac{3}{2}$

Explanation:

Step1: Split middle term

Find two numbers that multiply to $8\times(-9)=-72$ and add to $6$: $12$ and $-6$. Rewrite the equation:
$8x^2 + 12x - 6x - 9 = 0$

Step2: Group and factor pairs

Group first/second and third/fourth terms, factor GCFs:
$(8x^2 + 12x) + (-6x - 9) = 0$
$4x(2x + 3) - 3(2x + 3) = 0$

Step3: Factor out common binomial

Factor $(2x+3)$ from both terms:
$(2x + 3)(4x - 3) = 0$

Step4: Solve for $x$

Set each factor equal to 0 and solve:
$2x + 3 = 0 \implies x = -\frac{3}{2}$
$4x - 3 = 0 \implies x = \frac{3}{4}$

Answer:

D. $x = \frac{3}{4}, x = -\frac{3}{2}$