Track / Stage 4 / 106 of 340 #366 of 1964

Problem 366

AMC 12 late, AIME early

Algebra Difficulty 4.7 Multiple choice

1. Given $a=\frac{-\sqrt{6}+\sqrt{6-4 \sqrt{2}}}{2}$ . Then $a^{3}+$ $\sqrt{6} a^{2}+\sqrt{2} a+\sqrt{6}$ is:

Pick one

Official solution

-、1.D.
Notice that $a=\frac{-\sqrt{6}+\sqrt{\sqrt{6}^{2}-4 \times 1 \times \sqrt{2}}}{2 \times 1}$ , then $a$ is a root of the equation $x^{2}+\sqrt{6} x+\sqrt{2}=0$ .
Therefore, $a^{2}+\sqrt{6} a+\sqrt{2}=0$ .
Thus, $a^{3}+\sqrt{6} a^{2}+\sqrt{2} a+\sqrt{6}$
$=a\left(a^{2}+\sqrt{6} a+\sqrt{2}\right)+\sqrt{6}=\sqrt{6} .$

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