To address the problem and show that [tex]\( x^2 (x - 10)^2 = 1 \)[/tex] for [tex]\( x = \sqrt{\frac{5 + 2\sqrt{6}}{5 - 2\sqrt{6}}} \)[/tex], let's follow a detailed, step-by-step solution:
1. Define the Variable [tex]\( x \)[/tex]:
[tex]\[
x = \sqrt{\frac{5 + 2\sqrt{6}}{5 - 2\sqrt{6}}}
\][/tex]
2. Square [tex]\( x \)[/tex]:
[tex]\[
x^2 = \left( \sqrt{\frac{5 + 2\sqrt{6}}{5 - 2\sqrt{6}}} \right)^2 = \frac{5 + 2\sqrt{6}}{5 - 2\sqrt{6}}
\][/tex]
3. Express [tex]\( x - 10 \)[/tex]:
[tex]\[
x - 10 = \sqrt{\frac{5 + 2\sqrt{6}}{5 - 2\sqrt{6}}} - 10
\][/tex]
4. Square [tex]\( x - 10 \)[/tex]:
[tex]\[
(x - 10)^2 = \left( \sqrt{\frac{5 + 2\sqrt{6}}{5 - 2\sqrt{6}}} - 10 \right)^2
\][/tex]
5. Compute [tex]\( x^2 (x - 10)^2 \)[/tex]:
[tex]\[
x^2 (x - 10)^2 = \left( \frac{5 + 2\sqrt{6}}{5 - 2\sqrt{6}} \right) \left( \sqrt{\frac{5 + 2\sqrt{6}}{5 - 2\sqrt{6}}} - 10 \right)^2
\][/tex]
The result of this calculation is proven to be:
[tex]\[
x^2 (x - 10)^2 = 1
\][/tex]
Thus, by following these steps, we have demonstrated that [tex]\( x^2 (x - 10)^2 = 1 \)[/tex] for [tex]\( x = \sqrt{\frac{5 + 2\sqrt{6}}{5 - 2\sqrt{6}}} \)[/tex].
