Difference between revisions of "2018 AMC 10A Problems/Problem 10"
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− | + | We can substitute <math>a</math> for <math>25-x^2</math>, thus turning the equation into <math>\sqrt{a+24} - \sqrt{a} = 3</math>. Squaring gives us <math>a + 24 = 9 + 6\sqrt{a} + a</math>, solving for <math>a</math> gives us 25/4. We substitute this value into the expression they asked us to evaluate giving 8. | |
==Video Solution (HOW TO THINK CREATIVELY!)== | ==Video Solution (HOW TO THINK CREATIVELY!)== |
Revision as of 11:29, 19 June 2024
Contents
[hide]Problem
Suppose that real number satisfies What is the value of ?
Solution 1
We let ; in other words, we want to find . We know that Thus, .
~Technodoggo
Solution 2
Let , and . Solving for the constants in terms of x, a , and b, we get , and . Subtracting the second equation from the first gives us . Difference of squares gives us . Since we want to find , and we know , we get , so
~idk12345678
Solution 3
We can substitute for , thus turning the equation into . Squaring gives us , solving for gives us 25/4. We substitute this value into the expression they asked us to evaluate giving 8.
Video Solution (HOW TO THINK CREATIVELY!)
~Education, the Study of Everything
Video Solutions
Video Solution 1
https://youtu.be/ba6w1OhXqOQ?t=1403
~ pi_is_3.14
Video Solution 2
https://youtu.be/zQG70XKAdeA ~ North America Math Contest Go Go Go
Video Solution 3
Video Solution 4
~savannahsolver
See Also
2018 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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 |