To solve the quadratic equation [tex]\( x^2 + 8x = 9 \)[/tex] by completing the square, follow these steps:
1. Move the constant term to the right-hand side:
[tex]\[
x^2 + 8x = 9
\][/tex]
2. Add and subtract the same value to complete the square on the left-hand side. This value is [tex]\(\left( \frac{8}{2} \right)^2 = 16\)[/tex]:
[tex]\[
x^2 + 8x + 16 = 9 + 16
\][/tex]
3. Rewrite the left-hand side as a perfect square and simplify the right-hand side:
[tex]\[
(x + 4)^2 = 25
\][/tex]
4. Take the square root of both sides, remembering to consider both the positive and negative roots:
[tex]\[
x + 4 = \pm 5
\][/tex]
5. Solve for [tex]\(x\)[/tex] by isolating it:
For the positive root:
[tex]\[
x + 4 = 5
\][/tex]
[tex]\[
x = 5 - 4
\][/tex]
[tex]\[
x = 1
\][/tex]
For the negative root:
[tex]\[
x + 4 = -5
\][/tex]
[tex]\[
x = -5 - 4
\][/tex]
[tex]\[
x = -9
\][/tex]
6. Write the final solutions:
[tex]\[
x = -9, 1
\][/tex]
Thus, the solutions to the equation [tex]\( x^2 + 8x = 9 \)[/tex] are [tex]\( x = -9 \)[/tex] and [tex]\( x = 1 \)[/tex].
