Solve the equation by factoring.

9 x squared plus 21 x minus 8 equals 0

To solve the equation \( 9x^2 + 21x - 8 = 0 \) by factoring, follow these steps:

### Step 1: Identify the coefficients
The quadratic equation is in the form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 9 \)
- \( b = 21 \)
- \( c = -8 \)

### Step 2: Find two numbers whose product is \( ac \) and whose sum is \( b \)
We need two numbers whose product is \( a \times c = 9 \times (-8) = -72 \), and whose sum is \( b = 21 \).

Let's find the pair of numbers:
- The pair of numbers that multiply to give \( -72 \) and add up to \( 21 \) is \( 24 \) and \( -3 \).

### Step 3: Rewrite the middle term using the pair of numbers
We can rewrite the equation as:
\[
9x^2 + 24x - 3x - 8 = 0
\]

### Step 4: Factor by grouping
Now, group the terms:
\[
(9x^2 + 24x) - (3x + 8) = 0
\]

Factor each group:
\[
3x(3x + 8) - 1(3x + 8) = 0
\]

### Step 5: Factor out the common binomial factor
Factor out the common binomial factor \( (3x + 8) \):
\[
(3x + 8)(3x - 1) = 0
\]

### Step 6: Solve for \( x \)
Set each factor equal to zero and solve for \( x \):
1. \( 3x + 8 = 0 \)  
   \[
   3x = -8 \quad \Rightarrow \quad x = -\frac{8}{3}
   \]
2. \( 3x - 1 = 0 \)  
   \[
   3x = 1 \quad \Rightarrow \quad x = \frac{1}{3}
   \]

### Final Solutions:
The solutions to the equation are:
\[
x = -\frac{8}{3} \quad \text{or} \quad x = \frac{1}{3}
\]

Do you want more details or have any questions?

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Here are five related questions:
1. How do you solve a quadratic equation using the quadratic formula?
2. What is the significance of factoring in solving quadratic equations?
3. Can all quadratic equations be solved by factoring?
4. How does factoring differ from completing the square?
5. How can you check if your factorization is correct?

**Tip:** Always check your factorization by expanding the factors to ensure they match the original equation!

Solve the equation by factoring: 9x^2 + 21x - 8 = 0

Discover how to solve the quadratic equation 9x^2 + 21x - 8 by factoring. This detailed step-by-step solution covers key algebraic concepts such as factoring by grouping. Ideal for students in Grades 9-10.

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