Question

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

Answer 1

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

1. **Rearrange the equation**: First, move \( 9 \) to the left side of the equation.

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

2. **Isolate the quadratic terms**: Move the constant term to the other side.

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

3. **Complete the square**: Take the coefficient of \( x \) (which is \( 8 \)), divide it by \( 2 \) (giving \( 4 \)), and then square it (yielding \( 16 \)). Add this value to both sides of the equation.

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

Simplifying the right side:

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

4. **Rewrite the left side as a square**: The left side can be factored as:

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

5. **Take the square root of both sides**:

\[
x + 4 = \pm 5
\]

6. **Solve for \( x \)**: There are two cases to consider.

- **Case 1**:

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

- **Case 2**:

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

The two values that solve the equation \( x^2 + 8x = 9 \) are

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