Sure! Let’s solve the given equation step by step.
We start with the given equation:
[tex]\[ x - 4\sqrt{x} + 4 = 0 \][/tex]
(i) Let [tex]\( u = \sqrt{x} \)[/tex]. Then [tex]\( u^2 = x \)[/tex].
Substitute [tex]\( u \)[/tex] and [tex]\( u^2 \)[/tex] into the original equation:
[tex]\[ u^2 - 4u + 4 = 0 \][/tex]
Now, we need to solve this quadratic equation for [tex]\( u \)[/tex].
To solve the quadratic equation [tex]\( u^2 - 4u + 4 = 0 \)[/tex], we can use factoring.
Notice that the quadratic [tex]\( u^2 - 4u + 4 \)[/tex] can be written as:
[tex]\[ (u - 2)(u - 2) = 0 \][/tex]
or
[tex]\[ (u - 2)^2 = 0 \][/tex]
For the product to be zero, one of the factors must be zero:
[tex]\[ u - 2 = 0 \][/tex]
So,
[tex]\[ u = 2 \][/tex]
Therefore, the value of [tex]\( u \)[/tex] that satisfies the given equation is:
[tex]\[ u = 2 \][/tex]
Thus, we have shown that [tex]\( u = 2 \)[/tex].
