Sure! Let's solve the given equation step-by-step. The equation we have is:
[tex]\[
x^2 - 65x = 64 \sqrt{x}
\][/tex]
First, we aim to isolate [tex]\(x\)[/tex]. To do this more efficiently, we will use the quadratic equation:
[tex]\[x^2 - 65x - 64\sqrt{x} = 0\][/tex]
For simplicity, let us define [tex]\(y = \sqrt{x}\)[/tex]. Therefore, we have [tex]\(x = y^2\)[/tex], and substituting into the equation gives:
[tex]\[ (y^2)^2 - 65(y^2) - 64y = 0 \][/tex]
[tex]\[ y^4 - 65y^2 - 64y = 0 \][/tex]
This can be solved as a quadratic in terms of [tex]\(y^2\)[/tex]. This is not straightforward to solve without numerical computation, but knowing the solutions, we find:
[tex]\[y^4 - 65y^2 - 64y = 0\][/tex]
The roots discovered are:
[tex]\[ x = 0 \][/tex]
[tex]\[ x = \frac{\sqrt{257}}{2} + \frac{129}{2} \][/tex]
We disregard the [tex]\(0\)[/tex] solution since we are dealing with square roots, and we focus on the positive value [tex]\( x = \frac{\sqrt{257}}{2} + \frac{129}{2} \)[/tex].
Given [tex]\( x \)[/tex] as positive, we need to find:
[tex]\[
\sqrt{x - \sqrt{x}}
\][/tex]
Substitute the value [tex]\( x = \frac{\sqrt{257}}{2} + \frac{129}{2} \)[/tex]:
[tex]\[
\sqrt{\left( \frac{\sqrt{257}}{2} + \frac{129}{2} \right) - \sqrt{ \frac{\sqrt{257}}{2} + \frac{129}{2} }}
\][/tex]
This results in:
[tex]\[
\sqrt{ -\sqrt{\left( \frac{\sqrt{257}}{2} + \frac{129}{2} \right) } + \left( \frac{\sqrt{257}}{2} + \frac{129}{2} \right)}
\][/tex]
Thus, the final answer, simplifying and keeping it as it is since it involves nested radicals, remains:
[tex]\[
\sqrt{ -\sqrt{\left( \frac{\sqrt{257}}{2} + \frac{129}{2} \right)} + \left( \frac{\sqrt{257}}{2} + \frac{129}{2} \right)}
\][/tex]
This is the final simplified form for [tex]\(\sqrt{x - \sqrt{x}}\)[/tex].
