Question

solve the equation by factoring.
$9x^2 + 21x - 8 = 0$
the solution set is {}.
(type an integer or a simplified fraction. use a comma to se

Explanation:

Step1: Find two numbers for factoring

We need two numbers that multiply to \(9\times(-8)= -72\) and add up to \(21\). The numbers are \(24\) and \(-3\) since \(24\times(-3)= -72\) and \(24 + (-3)=21\).
Rewrite the middle term using these numbers:
\(9x^{2}+24x - 3x - 8 = 0\)

Step2: Group and factor

Group the first two terms and the last two terms:
\((9x^{2}+24x)+(-3x - 8)=0\)
Factor out the greatest common factor from each group:
\(3x(3x + 8)-1(3x + 8)=0\)

Step3: Factor out the common binomial

Factor out \((3x + 8)\):
\((3x + 8)(3x - 1)=0\)

Step4: Solve for \(x\)

Set each factor equal to zero and solve:

• For \(3x + 8 = 0\):

\(3x=-8\)
\(x = -\frac{8}{3}\)

• For \(3x - 1 = 0\):

\(3x = 1\)
\(x=\frac{1}{3}\)

Answer:

\(-\frac{8}{3},\frac{1}{3}\)