Sure, let's solve this step by step.
### 1) Rewrite the equation by completing the square
The original equation given is:
[tex]\[ x^2 + 8x + 7 \][/tex]
To complete the square, follow these steps:
1. Move the constant term to the right side:
[tex]\[ x^2 + 8x = -7 \][/tex]
2. Add and subtract the necessary term to complete the square on the left side:
- Take half the coefficient of [tex]\(x\)[/tex], which is [tex]\(8/2 = 4\)[/tex].
- Square it to get [tex]\(4^2 = 16\)[/tex].
- Add and subtract this square term on the left side of the equation:
[tex]\[ x^2 + 8x + 16 - 16 = -7 \][/tex]
3. Rewrite it in the form of a perfect square on the left side:
[tex]\[ (x + 4)^2 - 16 = -7 \][/tex]
4. Move the constant term ([tex]\(-16\)[/tex]) to the right side:
[tex]\[ (x + 4)^2 = 9 \][/tex]
So, the completed square form of the given equation is:
[tex]\[ 9 = (x + 4)^2 \][/tex]
### 2) What are the solutions to the equation?
Now, we solve the equation:
[tex]\[ (x + 4)^2 = 9 \][/tex]
Taking the square root of both sides, we get:
[tex]\[
x + 4 = \pm \sqrt{9}
\][/tex]
This gives us two separate equations to solve:
[tex]\[
x + 4 = 3 \quad \text{or} \quad x + 4 = -3
\][/tex]
Solving both:
1. [tex]\(x + 4 = 3\)[/tex]
[tex]\[
x = 3 - 4
\][/tex]
[tex]\[
x = -1
\][/tex]
2. [tex]\(x + 4 = -3\)[/tex]
[tex]\[
x = -3 - 4
\][/tex]
[tex]\[
x = -7
\][/tex]
So, the solutions to the equation are:
[tex]\[ x = -1 \quad \text{and} \quad x = -7 \][/tex]
However, if we consider the answer format provided, we summarize the steps as:
[tex]\[ x = -4 \pm 3 \][/tex]
Since one value is:
[tex]\[
-4 + 3 = -1
\][/tex]
And the other value is:
[tex]\[
-4 - 3 = -7
\][/tex]
### Final Answer
Choose the correct answer from the provided options:
(A) [tex]\(x = -4 \pm 3\)[/tex]
So the correct answer is:
[tex]\[
\boxed{x = -4 \pm 3}
\][/tex]
