Sure, let's solve the quadratic equation [tex]\( x^2 - 6x + 9 = 2 \)[/tex] by using the method of completing the square. We will also check the possible solutions against the given options.
### Step-by-Step Solution:
1. Original Equation:
[tex]\[ x^2 - 6x + 9 = 2 \][/tex]
2. Move the constant term to the right-hand side:
[tex]\[ x^2 - 6x + 9 - 2 = 0 \][/tex]
Simplify:
[tex]\[ x^2 - 6x + 7 = 0 \][/tex]
3. Rewrite the equation:
Notice that [tex]\(9\)[/tex] in the left-hand side is a perfect square. So, we start by isolating this term:
[tex]\[ x^2 - 6x + (9 - 7) = 0 \][/tex]
[tex]\[ x^2 - 6x + 2 = 0 \][/tex]
4. Complete the square:
We rewrite the quadratic equation in a form where we can complete the square. We start with the left-hand side part:
[tex]\[ x^2 - 6x + 9 - 9 + 7 = 0 \][/tex]
[tex]\[ (x^2 - 6x + 9) - 2 = 0 \][/tex]
[tex]\[ (x - 3)^2 - 2 = 0 \][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[ (x - 3)^2 = 2 \][/tex]
6. Take the square root of both sides:
[tex]\[ x - 3 = \pm \sqrt{2} \][/tex]
7. Solve for [tex]\( x \)[/tex]:
[tex]\[ x - 3 = \sqrt{2} \quad \text{or} \quad x - 3 = -\sqrt{2} \][/tex]
Therefore:
[tex]\[ x = 3 + \sqrt{2} \quad \text{or} \quad x = 3 - \sqrt{2} \][/tex]
### Verifying the Solutions with the Given Options:
- Option A: [tex]\( x = -3 - \sqrt{2} \)[/tex]
This does not match the solutions we found.
- Option B: [tex]\( x = 3 + \sqrt{2} \)[/tex]
This matches one of our solutions.
- Option C: [tex]\( x = -3 + \sqrt{2} \)[/tex]
This does not match the solutions we found.
- Option D: [tex]\( x = 3 - \sqrt{2} \)[/tex]
This matches one of our solutions.
Therefore, the correct options are:
- B. [tex]\( x = 3 + \sqrt{2} \)[/tex]
- D. [tex]\( x = 3 - \sqrt{2} \)[/tex]
