To solve the quadratic equation using the method of completing the square, follow these detailed steps:
1. Start with the original equation:
[tex]\[
x^2 + 6x + 9 = 2
\][/tex]
2. Move the constant term on the left side to the right side of the equation:
[tex]\[
x^2 + 6x + 9 - 2 = 0
\][/tex]
[tex]\[
x^2 + 6x + 7 = 0
\][/tex]
3. Complete the square:
- Take the coefficient of [tex]\(x\)[/tex], which is 6, divide it by 2, and then square it:
[tex]\[
\left(\frac{6}{2}\right)^2 = 3^2 = 9
\][/tex]
- Add and subtract this square inside the equation:
[tex]\[
x^2 + 6x + 9 - 9 + 7 = 0
\][/tex]
- Reformat the left side as a squared term and combine constants on the right side:
[tex]\[
(x + 3)^2 - 2 = 0
\][/tex]
4. Isolate the squared term:
[tex]\[
(x + 3)^2 = 2
\][/tex]
5. Take the square root of both sides:
[tex]\[
x + 3 = \pm\sqrt{2}
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
- For the positive square root:
[tex]\[
x + 3 = \sqrt{2}
\][/tex]
[tex]\[
x = \sqrt{2} - 3
\][/tex]
Which simplifies to:
[tex]\[
x = -3 + \sqrt{2}
\][/tex]
- For the negative square root:
[tex]\[
x + 3 = -\sqrt{2}
\][/tex]
[tex]\[
x = -\sqrt{2} - 3
\][/tex]
Which simplifies to:
[tex]\[
x = -3 - \sqrt{2}
\][/tex]
So, the solutions to the quadratic equation [tex]\(x^2 + 6x + 9 = 2\)[/tex] are:
[tex]\[
x = -3 + \sqrt{2} \quad \text{and} \quad x = -3 - \sqrt{2}
\][/tex]
Hence, the correct answers are:
- [tex]\(x = -3 - \sqrt{2}\)[/tex] (Option A)
- [tex]\(x = -3 + \sqrt{2}\)[/tex] (Option C)
