Question

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

Answer 1

To complete the square for the quadratic equation \( x^2 + 8x = 9 \), we follow these steps:

1. Move the constant to the other side of the equation:

\[
x^2 + 8x - 9 = 0
\]

2. Rearrange it slightly:

\[
x^2 + 8x = 9
\]

3. To complete the square, we need to 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. Now add 16 to both sides of the equation:

\[
x^2 + 8x + 16 = 9 + 16
\]

\[
x^2 + 8x + 16 = 25
\]

5. Now, the left side can be factored as a perfect square:

\[
(x + 4)^2 = 25
\]

6. To solve for \( x \), take the square root of both sides:

\[
x + 4 = \pm 5
\]

7. Now, solve for \( x \) in both cases:

- For the positive case:
\[
x + 4 = 5 \implies x = 5 - 4 \implies x = 1
\]

- For the negative case:
\[
x + 4 = -5 \implies x = -5 - 4 \implies x = -9
\]

Therefore, the two solutions to the equation \( x^2 + 8x = 9 \) are

\[
\boxed{1} \text{ and } \boxed{-9}.
\]