Question
Solve the equation by factoring. \[ 9 x^{2}+21 x-8=0 \] The solution set is \( \{\square\} \). (Use a comma to separate answers as needed. Type each solution only once.)
Ask by Salinas Huff. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions are \( x = \frac{1}{3} \) and \( x = -\frac{8}{3} \).
Solution
To solve the quadratic equation \( 9x^{2} + 21x - 8 = 0 \) by factoring, follow these steps:
1. **Find two numbers that multiply to \( 9 \times (-8) = -72 \) and add to \( 21 \).**
The numbers are **24** and **-3** because:
\[
24 \times (-3) = -72 \quad \text{and} \quad 24 + (-3) = 21
\]
2. **Rewrite the middle term using these numbers:**
\[
9x^{2} + 24x - 3x - 8 = 0
\]
3. **Factor by grouping:**
\[
(9x^{2} + 24x) + (-3x - 8) = 0
\]
\[
3x(3x + 8) -1(3x + 8) = 0
\]
\[
(3x - 1)(3x + 8) = 0
\]
4. **Solve for \( x \):**
\[
3x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{3}
\]
\[
3x + 8 = 0 \quad \Rightarrow \quad x = -\frac{8}{3}
\]
**Solution set:**
\[
\left\{ \frac{1}{3}, -\frac{8}{3} \right\}
\]
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution Bonus Knowledge
To solve the equation \( 9x^{2} + 21x - 8 = 0 \) by factoring, we start by finding two numbers that multiply to \( 9 \times -8 = -72 \) and add to \( 21 \). The numbers \( 24 \) and \( -3 \) do the trick. Next, we rewrite the expression: \[ 9x^{2} + 24x - 3x - 8 = 0 \] Now, we group and factor: \[ (9x^{2} + 24x) + (-3x - 8) = 0 \] Factoring gives us: \[ 3x(3x + 8) - 1(3x + 8) = 0 \] This can be factored further: \[ (3x - 1)(3x + 8) = 0 \] Setting each factor to zero: 1. \( 3x - 1 = 0 \) leads to \( x = \frac{1}{3} \) 2. \( 3x + 8 = 0 \) leads to \( x = -\frac{8}{3} \) Thus, the solution set is \( \left\{ \frac{1}{3}, -\frac{8}{3} \right\} \).
