Maths Olympiad Prep

Track / Stage 4 / 307 of 340 #567 of 1964

Problem 567

AMC 12 late, AIME early
Algebra Difficulty 5.0 Find the answer

G2.1 If a=x4+x−4a=x^{4}+x^{-4} and x2+x+1=0x^{2}+x+1=0, find the value of aa.

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

Official solution

x2+x+1x=0⇒x+1x=−1⇒(x+1x)2=1⇒x2+1x2=−1⇒(x2+1x2)2=1a=x4+1x4=−1\begin{array}{l}\frac{x^{2}+x+1}{x}=0 \\ \Rightarrow x+\frac{1}{x}=-1 \\ \Rightarrow\left(x+\frac{1}{x}\right)^{2}=1 \\ \Rightarrow x^{2}+\frac{1}{x^{2}}=-1 \\ \Rightarrow\left(x^{2}+\frac{1}{x^{2}}\right)^{2}=1 \\ a=x^{4}+\frac{1}{x^{4}}=-1\end{array}

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.