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Problem 527

AMC 12 late, AIME early

Algebra Difficulty 4.9 Find the answer

2.122 Given real numbers $a, b$ satisfy $\frac{4}{a^{4}}-\frac{2}{a^{2}}-3=0$ and $b^{4}+b^{2}-3=0$ . Then the value of the algebraic expression $\frac{a^{4} b^{4}+4}{a^{4}}$ is
(A) 175
(B) 55 .
(C) 13 .
(D) 7 .
(China Beijing Junior High School Mathematics Competition, 1990)

Your answer

Official solution

[Solution] From the given, we know $\left(\frac{2}{a^{2}}\right)^{2}-\left(\frac{2}{a^{2}}\right)-3=0$ , then $\frac{2}{a^{2}}=\frac{1+\sqrt{13}}{2}$ (negative value discarded).
Similarly, $b^{2}=\frac{-1+\sqrt{13}}{2}$ (negative value discarded). $\therefore \quad \frac{a^{4} b^{4}+4}{a^{4}}=b^{4}+\frac{4}{a^{4}}=\left(3-b^{2}\right)+\left(\frac{2}{a^{2}}+3\right)=7$ . Therefore, the answer is $(D)$ .

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