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Algebra Difficulty 4.2 AIME Find the answer Philippines

Problem:
Let xx be a real number satisfying x2−6x+1=0x^{2}-\sqrt{6} x+1=0. Find the numerical value of ∣x4−1x4∣\left|x^{4}-\frac{1}{x^{4}}\right|.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
Note that x+1x=6x+\frac{1}{x}=\sqrt{6}. Then
∣x4−1x4∣=(x2+1x2)∣x2−1x2∣=(x2+1x2)(x+1x)∣x−1x∣=((x+1x)2−2)(x+1x)((x+1x)2−4)=4(6)(2)=83 \begin{aligned} \left|x^{4}-\frac{1}{x^{4}}\right| &=\left(x^{2}+\frac{1}{x^{2}}\right)\left|x^{2}-\frac{1}{x^{2}}\right| \\ &=\left(x^{2}+\frac{1}{x^{2}}\right)\left(x+\frac{1}{x}\right)\left|x-\frac{1}{x}\right| \\ &=\left(\left(x+\frac{1}{x}\right)^{2}-2\right)\left(x+\frac{1}{x}\right)\left(\sqrt{\left(x+\frac{1}{x}\right)^{2}-4}\right) \\ &=4(\sqrt{6})(\sqrt{2})=8 \sqrt{3} \end{aligned}

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.