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
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.
However, this is not the final answer. Plug it back into the original equation to ensure it still works.
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 |