Difference between revisions of "2007 AMC 10A Problems/Problem 20"
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=== Solution 2(LIFEHACK) === | === Solution 2(LIFEHACK) === | ||
− | Notice that <math>(a^{4} + a^{-4}) = (a^{2} + a^{-2} + 2</math>. Since D is the only option 2 less than a perfect square, that is correct. | + | Notice that <math>(a^{4} + a^{-4}) = (a^{2} + a^{-2})^{2} + 2</math>. Since D is the only option 2 less than a perfect square, that is correct. |
PS: Because this is a multiple choice test, this works. | PS: Because this is a multiple choice test, this works. | ||
+ | |||
=== Solution 3 === | === Solution 3 === | ||
<math>4a = a^2 + 1</math>. We apply the [[quadratic formula]] to get <math>a = \frac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3}</math>. | <math>4a = a^2 + 1</math>. We apply the [[quadratic formula]] to get <math>a = \frac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3}</math>. |
Revision as of 22:14, 4 January 2020
Problem
Suppose that the number satisfies the equation . What is the value of ?
Solutions
Solution 1
Notice that . Thus .
Solution 2(LIFEHACK)
Notice that . Since D is the only option 2 less than a perfect square, that is correct.
PS: Because this is a multiple choice test, this works.
Solution 3
. We apply the quadratic formula to get .
Thus (so it doesn't matter which root of we use). Using the binomial theorem we can expand this out and collect terms to get .
Solution 3
(similar to Solution 1) We know that . We can square both sides to get , so . Squaring both sides again gives , so .
Solution 4
We let and be roots of a certain quadratic. Specifically . We use Newton's Sums given the coefficients to find .
Solution 5
Let = + . Then so . Then by De Moivre's Theorem, = and solving gets 194.
See also
2007 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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.