2011 AMC 10B Problems/Problem 19

Problem 19

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

$\textbf{(A)}\ -64 \qquad\textbf{(B)}\ -24 \qquad\textbf{(C)}\ -9 \qquad\textbf{(D)}\ 24 \qquad\textbf{(E)}\ 576$

Solution

First, square both sides, and isolate the absolute value.

\begin{align*}
5|x|+8&=x^2-16\\
5|x|&=x^2-24\\
|x|&=\frac{x^2-24}{5} (Error compiling LaTeX. Unknown error_msg)

Solve for the absolute value and factor. $x=\frac{x^2-24}{5} \longrightarrow 5x=x^2-24 \longrightarrow 0=x^2-5x-24 \longrightarrow (x-8)(x+3)=0\\ x=\frac{24-x^2}{5} \longrightarrow 5x=24-x^2 \longrightarrow 0=x^2+5x-24 \longrightarrow (x+8)(x-3)=0$ $x= -8, -3, 3, 8$

However, this is not the final answer. Plug it back into the original equation to ensure it still works. $\sqrt{5|-8|+8}=\sqrt{(-8)^2-16} \longrightarrow \sqrt{40+8}=\sqrt{64-16} \longrightarrow \sqrt{48}=\sqrt{48}\\ \sqrt{5|-3|+8}=\sqrt{(-3)^2-16} \longrightarrow \sqrt{15+8}=\sqrt{9-16} \longrightarrow \sqrt{23}\not=\sqrt{-5}\\ \sqrt{5|3|+8}=\sqrt{3^2-16} \longrightarrow\sqrt{15+8}=\sqrt{9-16} \longrightarrow \sqrt{23}\not=\sqrt{-5-16}\\ \sqrt{5|8|+8}=\sqrt{8^2-16} \longrightarrow \sqrt{40+8}=\sqrt{64-16} \longrightarrow \sqrt{48}=\sqrt{48}$ The roots of this equation are $-8$ and $8$ and product is $-8 \times 8 = \boxed{\textbf{(A)} -64}$

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2011 AMC 10B Problems/Problem 19

Problem 19

Solution