Question

Complete the square to identify the two values that solve the following quadratic equation x^2+8x = 9

Answer 1

To solve the quadratic equation \( x^2 + 8x = 9 \) by completing the square, follow these steps:

1. **Move 9 to the left side of the equation**:
\[
x^2 + 8x - 9 = 0
\]

2. **Rearrange to isolate the quadratic and linear terms**:
\[
x^2 + 8x = 9
\]

3. **Complete the square**:
To complete the square for \( x^2 + 8x \), take half of the coefficient of \( x \) (which is 8), square it, and add it to both sides. Half of 8 is 4, and squaring it gives \( 4^2 = 16 \).

Add 16 to both sides:
\[
x^2 + 8x + 16 = 9 + 16
\]
Simplifying gives:
\[
x^2 + 8x + 16 = 25
\]

4. **Factor the left side**:
The left side can be rewritten as a squared term:
\[
(x + 4)^2 = 25
\]

5. **Take the square root of both sides**:
\[
x + 4 = \pm 5
\]

6. **Solve for \( x \)**:
Now, we have two cases to solve:
- Case 1: \( x + 4 = 5 \)
\[
x = 5 - 4 = 1
\]
- Case 2: \( x + 4 = -5 \)
\[
x = -5 - 4 = -9
\]

Thus, the two values that solve the equation \( x^2 + 8x = 9 \) are:
\[
\boxed{1} \quad \text{and} \quad \boxed{-9}
\]