Question

Complete the square to identify the two values that solve the following quadratic equation: x2+8x=9 .

Answer 1

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

1. **Rearrange the equation**: Move the constant term to the left side.

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

2. **Isolate the constant term**: We want the left side to represent a perfect square.

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

3. **Complete the square**: Take the coefficient of \( x \) (which is 8), divide it by 2 (giving us 4), and square it (resulting in 16). Now, add and subtract this square inside the equation.

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

This can be rewritten as:

\[
(x + 4)^2 - 16 = 9
\]

4. **Rearrange it to a standard form**:

\[
(x + 4)^2 = 9 + 16
\]

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

5. **Take the square root of both sides**: Remember to consider both the positive and negative roots.

\[
x + 4 = 5 \quad \text{or} \quad x + 4 = -5
\]

6. **Solve for \( x \)**:

- For \( x + 4 = 5 \):

\[
x = 5 - 4 = 1
\]

- For \( x + 4 = -5 \):

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

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

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