Question

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

x=___

Answer 1

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

1. Move the constant to the other side of the equation:
\[
x^2 + 8x - 9 = 0
\]

2. Rewrite \( x^2 + 8x \) in a way that allows us to complete the square. First, take half of the coefficient of \( x \) (which is 8), square it, and add it inside the equation:
- Half of 8 is 4, and squaring it gives us \( 4^2 = 16 \).

We can rewrite the equation by adding and subtracting 16 on the left side:
\[
x^2 + 8x + 16 - 16 = 9
\]

This simplifies to:
\[
(x + 4)^2 - 16 = 9
\]

3. Move the constant (-16) to the right side of the equation:
\[
(x + 4)^2 = 9 + 16
\]
\[
(x + 4)^2 = 25
\]

4. Take the square root of both sides:
\[
x + 4 = \pm 5
\]

5. Solve for \( x \):
- For the positive case:
\[
x + 4 = 5 \implies x = 5 - 4 = 1
\]
- For the negative case:
\[
x + 4 = -5 \implies x = -5 - 4 = -9
\]

Thus, the two values that solve the equation are:
\[
x = 1 \quad \text{and} \quad x = -9
\]

Finally, to fill in the blank:
x=____ (you can write either 1 or -9 depending on the requirements for the answer)