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

Below is the professionally curated solution for Problem 19 of the 2011 AMC 10B, 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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Concepts:radical absolute value quadratic

Difficulty rating: 1580

19.

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

$-64$

$-24$

$-9$

$24$

$576$

Solution:

The equation is equal to $\sqrt{5 | x | + 8} = \sqrt{|x|^2 - 16}.$ Solving, we get that: $\begin{align*} 5|x|+8 &= |x|^2-16 \\ |x|^2-5|x|-24&=0 \\ (|x|-8)(|x|+3)&=0 \end{align*}$ This makes $|x|=-3,8,$ making $|x|=8$ the only possible value. Thus, $x =8,-8$ with a product of $-64.$

Thus, the correct answer is A .

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