What is the product of all the roots of the equation √5 | x | + 8 = √x^2 - 16?

2011 AMC 10B Problem 19 — Solution

2011 AMC 10B Problem 19

Attempt Problem 19 of the 2011 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 10B solutions, or check the answer key.

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19.

What is the product of all the roots of the equation $\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}?$

$-64$

$-24$

$-9$

$24$

$576$

Answer: A

Concepts:radical absolute value quadratic

Difficulty rating: 1580

Solution:

Because $x^2=|x|^2,$ squaring the equation gives $\begin{aligned} 5|x|+8 &= |x|^2-16,\\ |x|^2-5|x|-24&=0,\\ (|x|-8)(|x|+3)&=0. \end{aligned}$ The algebraic possibilities for $|x|$ are $8$ and $-3,$ but an absolute value cannot be negative. Hence $|x|=8,$ so the roots are $8$ and $-8$ , with product $-64.$

Thus, the correct answer is A .

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