To solve the equation [tex]\( x^2 = 12x - 15 \)[/tex] by completing the square, follow these steps:
1. Rewrite the equation:
[tex]\[
x^2 = 12x - 15
\][/tex]
Move all terms to one side to set the equation to zero:
[tex]\[
x^2 - 12x + 15 = 0
\][/tex]
2. Rearrange the terms to prepare for completing the square. Focus on the quadratic and linear terms:
[tex]\[
x^2 - 12x = 15
\][/tex]
3. Complete the square:
Add and subtract the square of half the coefficient of [tex]\(x\)[/tex] (which is 12/2 = 6) inside the equation:
[tex]\[
x^2 - 12x + 36 = 15 + 36
\][/tex]
The left-hand side becomes a perfect square trinomial:
[tex]\[
(x - 6)^2 = 21
\][/tex]
4. Solve for [tex]\(x\)[/tex] by taking the square root of both sides:
[tex]\[
x - 6 = \pm \sqrt{21}
\][/tex]
5. Isolate [tex]\(x\)[/tex]:
[tex]\[
x = 6 \pm \sqrt{21}
\][/tex]
This means the solution set is:
[tex]\[
\{ 6 - \sqrt{21}, 6 + \sqrt{21} \}
\][/tex]
Comparing with the given choices:
[tex]\[
\{6-\sqrt{21}, 6+\sqrt{21}\}
\][/tex]
Therefore, the solution set of the equation [tex]\( x^2 = 12x - 15 \)[/tex] is:
[tex]\[
\{6 - \sqrt{21}, 6 + \sqrt{21}\}
\][/tex]
