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Difference between revisions of "2011 AMC 10B Problems/Problem 19"

m
(Some of the LaTeX for the equations was hard to read. Added a bit more annotation also.)
Line 9: Line 9:
 
First, square both sides, and isolate the absolute value.
 
First, square both sides, and isolate the absolute value.
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
−
5|x|+8&=x^2-16\\
+
5|x|+8&=x^2-16,\\
−
5|x|&=x^2-24\\
+
5|x|&=x^2-24,\\
−
|x|&=\frac{x^2-24}{5}
+
|x|&=\frac{x^2-24}{5}.
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 
Solve for the absolute value and factor.
 
Solve for the absolute value and factor.
−
<cmath>x=\frac{x^2-24}{5} \longrightarrow 5x=x^2-24 \longrightarrow 0=x^2-5x-24 \longrightarrow (x-8)(x+3)=0 \; \|||\ \;
+
[i] Case 1: <math>x=\frac{x^2-24}{5}</math> [/i]
−
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</cmath>
+
Multiplying both sides by <math>5</math> gives us
 +
<cmath> 5x=x^2-24.</cmath>
 +
Rearranging and factoring, we have
 +
<math>\begin{align}
 +
x^2-5x-24 &=0, \\
 +
(x-8)(x-3) &= 0.
 +
\end{align*}</math>
  
−
However, this is not the final answer. Plug it back into the original equation to ensure it still works. Whether the number is positive or negative does not matter since the absolute value or square will cancel it out anyways.
+
[i] Case 2: <math>x=\frac{-x^2+24}{5}</math> [/i]
−
<cmath>\sqrt{5|3|+8}=\sqrt{3^2-16} \longrightarrow\sqrt{15+8}=\sqrt{9-16} \longrightarrow \sqrt{23}\not=\sqrt{-7}\\
+
As above, we multiply both sides by <math>5</math> to find
−
\sqrt{5|8|+8}=\sqrt{8^2-16} \longrightarrow \sqrt{40+8}=\sqrt{64-16} \longrightarrow \sqrt{48}=\sqrt{48}</cmath>
+
<cmath> 5x=-x^2+24.</cmath>
−
The roots of this equation are <math>-8</math> and <math>8</math> and product is <math>-8 \times 8 = \boxed{\textbf{(A)} -64}</math>
+
Rearranging and factoring gives us
 +
<math>\begin{align}
 +
x^2+5x-24 &=0, \\
 +
(x+8)(x-3) &= 0.
 +
\end{align*}</math>
 +
 
 +
Combining these cases, we have <math>x= -8, -3, 3, 8</math>. Because our first step of squaring is not reversible, however, we need to check for extraneous solutions. Plug each solution for <math>x</math> back into the original equation to ensure it works. Whether the number is positive or negative does not matter since the absolute value or square will cancel it out anyways.
 +
Trying <math>|x|=|3|</math>, we have
 +
<math>\begin{align*}
 +
\sqrt{5|3|+8}&=\sqrt{3^2-16}, \\
 +
\sqrt{15+8}&=\sqrt{9-16}, \\
 +
\sqrt{23}\neq\sqrt{-7}.
 +
\end{align*}</math>
 +
Therefore, <math>x = 3</math> and <math> x= -3</math> are extraneous.
 +
 
 +
Checking <math>|x|=|8|</math>, we have
 +
<math>\begin{align*}
 +
\sqrt{5|8|+8}&=\sqrt{8^2-16}, \\
 +
\sqrt{40+8}&=\sqrt{64-16}, \\
 +
\sqrt{48}&=\sqrt{48}.
 +
\end{align*}</math>
 +
 
 +
The roots of our original equation are <math>-8</math> and <math>8</math> and product is <math>-8 \times 8 = \boxed{\textbf{(A)} -64}</math>.
  
 
== Solution 2 ==  
 
== Solution 2 ==  

Revision as of 20:35, 20 November 2020

Problem

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 1

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}. \end{align*} Solve for the absolute value and factor. [i] Case 1: $x=\frac{x^2-24}{5}$ [/i] Multiplying both sides by $5$ gives us \[5x=x^2-24.\] Rearranging and factoring, we have $\begin{align} x^2-5x-24 &=0, \\ (x-8)(x-3) &= 0. \end{align*}$ (Error compiling LaTeX. Unknown error_msg)

[i] Case 2: $x=\frac{-x^2+24}{5}$ [/i] As above, we multiply both sides by $5$ to find \[5x=-x^2+24.\] Rearranging and factoring gives us $\begin{align} x^2+5x-24 &=0, \\ (x+8)(x-3) &= 0. \end{align*}$ (Error compiling LaTeX. Unknown error_msg)

Combining these cases, we have $x= -8, -3, 3, 8$. Because our first step of squaring is not reversible, however, we need to check for extraneous solutions. Plug each solution for $x$ back into the original equation to ensure it works. Whether the number is positive or negative does not matter since the absolute value or square will cancel it out anyways. Trying $|x|=|3|$, we have $\begin{align*} \sqrt{5|3|+8}&=\sqrt{3^2-16}, \\ \sqrt{15+8}&=\sqrt{9-16}, \\ \sqrt{23}\neq\sqrt{-7}. \end{align*}$ (Error compiling LaTeX. Unknown error_msg) Therefore, $x = 3$ and $x= -3$ are extraneous.

Checking $|x|=|8|$, we have $\begin{align*} \sqrt{5|8|+8}&=\sqrt{8^2-16}, \\ \sqrt{40+8}&=\sqrt{64-16}, \\ \sqrt{48}&=\sqrt{48}. \end{align*}$ (Error compiling LaTeX. Unknown error_msg)

The roots of our original equation are $-8$ and $8$ and product is $-8 \times 8 = \boxed{\textbf{(A)} -64}$.

Solution 2

Square both sides, to get $5|x| + 8 = x^2-16$. Rearrange to get $x^2 - 5|x| - 24 = 0$. Seeing that $x^2 = |x|^2$, substitute to get $|x|^2 - 5|x| - 24 = 0$. We see that this is a quadratic in $|x|$. Factoring, we get $(|x|-8)(|x|+3) = 0$, so $|x| = \{8,-3\}$. Since the radicand of the equation can't be negative, the sole solution is $|x| = 8$. Therefore, the x can be $8$ or $-8$. The product is then $-8 \times 8 = \boxed{\textbf{(A)} -64}$.

See Also

2011 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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