Question

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

Answer 1

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

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

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

2. **Isolate the quadratic and linear terms:**

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

3. **Complete the square for the left-hand side. To do this, take half of the coefficient of \(x\) (which is 8), square it, and add it to both sides of the equation.**

Half of 8 is 4, and squaring it gives:

\[
4^2 = 16
\]

Now add 16 to both sides:

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

Simplifying the right-hand side gives:

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

4. **Rewrite the left-hand side as a square:**

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

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

\[
x + 4 = \pm 5
\]

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

- When \(x + 4 = 5\):
\[
x = 5 - 4 = 1
\]

- When \(x + 4 = -5\):
\[
x = -5 - 4 = -9
\]

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

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