Library / Philippine Mathematical Olympiad /4 of 114

Philippine Mathematical Olympiad

Algebra Difficulty 4.2 AIME Find the answer Philippines

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

Your answer

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

Solution

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

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

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