To solve the equation [tex]\( x - 12 \sqrt{x} + 36 = 0 \)[/tex], we need to find the value of [tex]\( x \)[/tex] that satisfies it. Let's proceed with the solution step-by-step.
Consider the equation:
[tex]\[ x - 12 \sqrt{x} + 36 = 0 \][/tex]
We can make a substitution to simplify the equation. Let [tex]\( y = \sqrt{x} \)[/tex]. Then [tex]\( x = y^2 \)[/tex]. Substituting these into the equation gives us:
[tex]\[ y^2 - 12y + 36 = 0 \][/tex]
This is a quadratic equation in terms of [tex]\( y \)[/tex]. We 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].
First, we calculate the discriminant:
[tex]\[ b^2 - 4ac = (-12)^2 - 4(1)(36) = 144 - 144 = 0 \][/tex]
Since the discriminant is zero, there is exactly one real solution:
[tex]\[ y = \frac{-b}{2a} = \frac{12}{2} = 6 \][/tex]
So, [tex]\( y = 6 \)[/tex]. Recall that we made the substitution [tex]\( y = \sqrt{x} \)[/tex]. Thus:
[tex]\[ \sqrt{x} = 6 \][/tex]
To find [tex]\( x \)[/tex], we square both sides:
[tex]\[ x = 6^2 = 36 \][/tex]
Now we check the given choices to select the correct option:
A. 6
B. [tex]\( 6^2 \)[/tex]
C. [tex]\( 6^3 \)[/tex]
D. [tex]\( 6^4 \)[/tex]
The value of [tex]\( x \)[/tex] is 36, which corresponds to [tex]\( 6^2 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
