Question

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

Answer 1

it is a one answer thing

Answer 2

To complete the square for the equation \( x^2 + 8x = 9 \), we start by rearranging the equation:

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

2. To complete the square, we need to take the coefficient of \( x \) (which is 8), divide it by 2 (getting 4), and then square that value:
\[
\left( \frac{8}{2} \right)^2 = 4^2 = 16
\]

3. Now, add and subtract this value on the left side of the equation:
\[
x^2 + 8x + 16 - 16 - 9 = 0
\]

4. Simplify the left side:
\[
(x + 4)^2 - 25 = 0
\]

5. Move the constant term to the right side:
\[
(x + 4)^2 = 25
\]

6. Take the square root of both sides, remembering to consider both the positive and negative roots:
\[
x + 4 = 5 \quad \text{or} \quad x + 4 = -5
\]

7. Solving these two equations gives us:
\[
x + 4 = 5 \quad \Rightarrow \quad x = 5 - 4 = 1
\]
\[
x + 4 = -5 \quad \Rightarrow \quad x = -5 - 4 = -9
\]

Thus, the two values that solve the equation \( x^2 + 8x = 9 \) are:
\[
\boxed{1} \quad \text{and} \quad \boxed{-9}.
\]

Answer 3

I understand you're looking for a singular solution based on the completed square method. Let's summarize the solution clearly in one concise form:

Starting with the equation:

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

We complete the square:

1. Move 9 to the left:
\[
x^2 + 8x - 9 = 0
\]

2. Complete the square:
- Take half of 8 (which is 4) and square it (which is 16).
- Rewrite:
\[
(x + 4)^2 - 16 - 9 = 0
\]
\[
(x + 4)^2 - 25 = 0
\]
\[
(x + 4)^2 = 25
\]

3. Take the square root:
\[
x + 4 = 5 \quad \text{or} \quad x + 4 = -5
\]

4. Solve for \( x \):
- From \( x + 4 = 5 \):
\[
x = 1
\]

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

Given you're looking for one answer, the solutions to the equation are \( x = 1 \) and \( x = -9 \).

If you're looking for a singular representation of the basic solution form, here it is:

**The solutions are: \(\{1, -9\}\).**