Question

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

Answer 1

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

1. Rearrange the equation to set it to zero:
\[
x^2 + 8x - 9 = 0
\]

2. Move the constant term to the other side:
\[
x^2 + 8x = 9
\]

3. To complete the square, 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:
\[
4^2 = 16
\]

4. Add 16 to both sides of the equation:
\[
x^2 + 8x + 16 = 9 + 16
\]
\[
x^2 + 8x + 16 = 25
\]

5. Now, the left side is a perfect square:
\[
(x + 4)^2 = 25
\]

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

7. Solve for \( x \):
- \( x + 4 = 5 \):
\[
x = 5 - 4 = 1
\]
- \( x + 4 = -5 \):
\[
x = -5 - 4 = -9
\]

Thus, the two values that solve the quadratic equation are:
\[
x = 1 \quad \text{and} \quad x = -9
\]

So, the answer is:
\[
x = 1, -9
\]