Let's solve the equation [tex]\(x - 12 \sqrt{x} + 36 = 0\)[/tex] step by step to find the value of [tex]\(x\)[/tex].
1. Start by setting the equation equal to zero:
[tex]\[
x - 12 \sqrt{x} + 36 = 0
\][/tex]
2. To simplify, let's make a substitution. Let [tex]\( y = \sqrt{x} \)[/tex]. Therefore, [tex]\( y^2 = x \)[/tex].
3. Substituting [tex]\( y \)[/tex] into the equation, we get:
[tex]\[
y^2 - 12y + 36 = 0
\][/tex]
4. This is a quadratic equation in [tex]\( y \)[/tex]. We can solve it using the quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -12 \)[/tex], and [tex]\( c = 36 \)[/tex].
5. Let's apply the quadratic formula:
[tex]\[
y = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 1 \cdot 36}}{2 \cdot 1}
\][/tex]
Simplify inside the square root:
[tex]\[
y = \frac{12 \pm \sqrt{144 - 144}}{2}
\][/tex]
[tex]\[
y = \frac{12 \pm \sqrt{0}}{2}
\][/tex]
[tex]\[
y = \frac{12 \pm 0}{2}
\][/tex]
[tex]\[
y = \frac{12}{2}
\][/tex]
[tex]\[
y = 6
\][/tex]
6. Recall that [tex]\( y = \sqrt{x} \)[/tex]. Substituting back, we have:
[tex]\[
\sqrt{x} = 6
\][/tex]
7. Square both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 6^2
\][/tex]
[tex]\[
x = 36
\][/tex]
So, the value of [tex]\( x \)[/tex] is [tex]\( 36 \)[/tex]. Therefore, the correct answer is:
[tex]\[
\boxed{6^2}
\][/tex]
