Question

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

Answer 1

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

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

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

2. Next, we'll complete the square on the left-hand side. 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 inside the equation:

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

Simplifying, we should move 16 to the other side:

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

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

4. Now, rewrite the left-hand side as a square:

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

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

\[
x + 4 = \pm 5
\]

6. Solve for \( x \) by isolating it:

- For the positive:

\[
x + 4 = 5 \implies x = 5 - 4 \implies x = 1
\]

- For the negative:

\[
x + 4 = -5 \implies x = -5 - 4 \implies x = -9
\]

The solutions to the quadratic equation \( x^2 + 8x = 9 \) are:

\[
x = 1 \quad \text{and} \quad x = -9
\]