2022 AMC 10A Problems/Problem 11
Contents
Problem
Ted mistakenly wrote as What is the sum of all real numbers for which these two expressions have the same value?
Solution 1
We are given that Converting everything into powers of we have We multiply both sides by , then rearrange as By Vieta's Formulas, the sum of such values of is
Note that or from the quadratic equation above.
~MRENTHUSIASM
~KingRavi
Solution 2 (Logarithms)
We can rewrite the equation using fractional exponents and take logarithms of both sides: We can then use the additive properties of logarithms to split them up: Using the power rule, the fact that and bringing the exponents down, we get from which or . Therefore, the answer is
- youtube.com/indianmathguy
Solution 3
Since surd roots are conventionally positive integers, assume is an integer, so can only be , , , , , and . . Testing out , we see that only and work. Hence, .
~MrThinker
Video Solution 1
~Education, the Study of Everything
Video Solution 2
Video Solution 3
~Whiz
Video Solution by TheBeautyofMath
~IceMatrix
See Also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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All AMC 10 Problems and Solutions |
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