To solve the equation [tex]\( x - 4 \sqrt{x} + 4 = 0 \)[/tex], let's proceed step by step.
1. Introduce a substitution:
Let [tex]\( y = \sqrt{x} \)[/tex]. This implies that [tex]\( y^2 = x \)[/tex].
2. Rewrite the equation using the substitution:
Substituting [tex]\( y \)[/tex] into the equation, we have:
[tex]\[
y^2 - 4y + 4 = 0
\][/tex]
3. Solve the quadratic equation:
The equation [tex]\( y^2 - 4y + 4 = 0 \)[/tex] is a standard quadratic equation and needs to be factored or solved using the quadratic formula. Notice that this equation can be factored easily:
[tex]\[
(y - 2)^2 = 0
\][/tex]
4. Find the roots of the factored equation:
Setting the factored form equal to zero gives us:
[tex]\[
(y - 2)^2 = 0 \implies y - 2 = 0 \implies y = 2
\][/tex]
5. Substitute back to find [tex]\( x \)[/tex]:
Recall our substitution [tex]\( y = \sqrt{x} \)[/tex]. Therefore:
[tex]\[
\sqrt{x} = 2
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Squaring both sides to eliminate the square root yields:
[tex]\[
x = 2^2 = 4
\][/tex]
Therefore, the solution to the equation [tex]\( x - 4 \sqrt{x} + 4 = 0 \)[/tex] is [tex]\(\boxed{4}\)[/tex].
