To solve the equation [tex]\(x^2 + 6x = 7\)[/tex] by completing the square, follow these steps:
### Step 1: Move the constant term to the other side of the equation
First, rewrite the given equation in the form [tex]\(x^2 + 6x - 7 = 0\)[/tex]:
[tex]\[x^2 + 6x = 7\][/tex]
### Step 2: Complete the square
To complete the square, we need to transform the left-hand side of the equation into a perfect square trinomial.
1. Take the coefficient of [tex]\(x\)[/tex], divide it by 2, and square it:
[tex]\[ \left(\frac{6}{2}\right)^2 = 3^2 = 9 \][/tex]
2. Add and subtract this square inside the equation:
[tex]\[ x^2 + 6x + 9 - 9 = 7 \][/tex]
3. Rewrite the equation as a perfect square trinomial:
[tex]\[ (x + 3)^2 - 9 = 7 \][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
1. Move the constant term [tex]\(-9\)[/tex] to the other side of the equation:
[tex]\[ (x + 3)^2 = 7 + 9 \][/tex]
[tex]\[ (x + 3)^2 = 16 \][/tex]
2. Take the square root of both sides:
[tex]\[ x + 3 = \pm 4 \][/tex]
3. Solve for [tex]\(x\)[/tex] by isolating it:
[tex]\[ x + 3 = 4 \quad \text{or} \quad x + 3 = -4 \][/tex]
4. Therefore, we have:
[tex]\[ x = 4 - 3 = 1 \][/tex]
[tex]\[ x = -4 - 3 = -7 \][/tex]
### Solution Set
The solution set of the equation [tex]\(x^2 + 6x = 7\)[/tex] is [tex]\(\{1, -7\}\)[/tex].
So, the correct answer is:
[tex]\[\boxed{\{-7, 1\}}\][/tex]
