Difference between revisions of "2021 AMC 10B Problems/Problem 2"
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==Problem== | ==Problem== | ||
− | What is the value of < | + | What is the value of <math>\sqrt{\left(3-2\sqrt{3}\right)^2}+\sqrt{\left(3+2\sqrt{3}\right)^2}</math>? |
<math>\textbf{(A)} ~0 \qquad\textbf{(B)} ~4\sqrt{3}-6 \qquad\textbf{(C)} ~6 \qquad\textbf{(D)} ~4\sqrt{3} \qquad\textbf{(E)} ~4\sqrt{3}+6</math> | <math>\textbf{(A)} ~0 \qquad\textbf{(B)} ~4\sqrt{3}-6 \qquad\textbf{(C)} ~6 \qquad\textbf{(D)} ~4\sqrt{3} \qquad\textbf{(E)} ~4\sqrt{3}+6</math> | ||
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~Interstigation | ~Interstigation | ||
+ | |||
+ | ==See Also== | ||
{{AMC10 box|year=2021|ab=B|num-b=1|num-a=3}} | {{AMC10 box|year=2021|ab=B|num-b=1|num-a=3}} | ||
+ | {{MAA Notice}} |
Latest revision as of 14:20, 2 March 2021
Contents
Problem
What is the value of ?
Solution
Note that the square root of any square is always the absolute value of the squared number because the square root function will only return a positive number. By squaring both and , we see that , thus is negative, so we must take the absolute value of , which is just . Knowing this, the first term in the expression equals and the second term is , and summing the two gives .
~bjc, abhinavg0627 and JackBocresion
Solution 2
Let , then . The term is there due to difference of squares. Simplifying the expression gives us , so ~ shrungpatel
Video Solution
https://youtu.be/HHVdPTLQsLc ~Math Python
Video Solution by OmegaLearn
Video Solution 3
~savannahsolver
Video Solution by TheBeautyofMath
https://youtu.be/gLahuINjRzU?t=154
~IceMatrix
Video Solution by Interstigation
https://youtu.be/DvpN56Ob6Zw?t=101
~Interstigation
See Also
2021 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.