Difference between revisions of "2011 AMC 10B Problems/Problem 19"
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| − | == Problem | + | == Problem== |
What is the product of all the roots of the equation <cmath>\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}.</cmath> | What is the product of all the roots of the equation <cmath>\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}.</cmath> | ||
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\sqrt{5|8|+8}=\sqrt{8^2-16} \longrightarrow \sqrt{40+8}=\sqrt{64-16} \longrightarrow \sqrt{48}=\sqrt{48}</cmath> | \sqrt{5|8|+8}=\sqrt{8^2-16} \longrightarrow \sqrt{40+8}=\sqrt{64-16} \longrightarrow \sqrt{48}=\sqrt{48}</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> | 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> | ||
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| + | == See Also== | ||
| + | |||
| + | {{AMC10 box|year=2011|ab=B|num-b=18|num-a=20}} | ||
Revision as of 19:50, 4 June 2011
Problem
What is the product of all the roots of the equation ![\[\sqrt{5 | x | + 8} = \sqrt{x^2 - 16}.\]](http://latex.artofproblemsolving.com/4/d/c/4dc3a78877196f453f6485a158c37ac0179def64.png)
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.
![\[x=\frac{x^2-24}{5} \longrightarrow 5x=x^2-24 \longrightarrow 0=x^2-5x-24 \longrightarrow (x-8)(x+3)=0\\ x=\frac{24-x^2}{5} \longrightarrow 5x=24-x^2 \longrightarrow 0=x^2+5x-24 \longrightarrow (x+8)(x-3)=0\]](http://latex.artofproblemsolving.com/d/6/8/d6830f2f96346ed1cd7f9ae55711126d9ace44af.png)
![\[x= -8, -3, 3, 8\]](http://latex.artofproblemsolving.com/2/b/c/2bc7598b9a1202046784f5096580e2bcea90bc31.png)
However, this is not the final answer. Plug it back into the original equation to ensure it still works.
![\[\sqrt{5|-8|+8}=\sqrt{(-8)^2-16} \longrightarrow \sqrt{40+8}=\sqrt{64-16} \longrightarrow \sqrt{48}=\sqrt{48}\\ \sqrt{5|-3|+8}=\sqrt{(-3)^2-16} \longrightarrow \sqrt{15+8}=\sqrt{9-16} \longrightarrow \sqrt{23}\not=\sqrt{-5}\\ \sqrt{5|3|+8}=\sqrt{3^2-16} \longrightarrow\sqrt{15+8}=\sqrt{9-16} \longrightarrow \sqrt{23}\not=\sqrt{-5-16}\\ \sqrt{5|8|+8}=\sqrt{8^2-16} \longrightarrow \sqrt{40+8}=\sqrt{64-16} \longrightarrow \sqrt{48}=\sqrt{48}\]](http://latex.artofproblemsolving.com/7/3/c/73cf7019ace670bc9f05a786a0d80671829289eb.png)
The roots of this equation are 
and 
and product is 
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 | ||
