To solve the quadratic equation \(x^2 + 8x = 9\) by completing the square, follow these steps:
1. Move the constant term to the other side of the equation:
\[
x^2 + 8x - 9 = 0
\]
or, equivalently,
\[
x^2 + 8x = 9.
\]
2. To complete the square, take the coefficient of \(x\) (which is 8), divide it by 2, and square it:
\[
\left(\frac{8}{2}\right)^2 = 4^2 = 16.
\]
3. Add and subtract this square (16) on the left side of the equation:
\[
x^2 + 8x + 16 - 16 = 9,
\]
or rewriting this gives:
\[
(x + 4)^2 - 16 = 9.
\]
4. Now, rearrange the equation:
\[
(x + 4)^2 = 9 + 16,
\]
which simplifies to:
\[
(x + 4)^2 = 25.
\]
5. Take the square root of both sides:
\[
x + 4 = \pm 5.
\]
6. Now, solve for \(x\) by isolating it in both cases:
\[
x + 4 = 5 \quad \Rightarrow \quad x = 5 - 4 = 1,
\]
and
\[
x + 4 = -5 \quad \Rightarrow \quad x = -5 - 4 = -9.
\]
Thus, the two values that solve the quadratic equation \(x^2 + 8x = 9\) are:
\[
x = 1 \quad \text{and} \quad x = -9.
\]