Difference between revisions of "2007 AMC 10A Problems/Problem 20"
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== Solution 1 (Decreases the Powers) == | == Solution 1 (Decreases the Powers) == | ||
− | Note that for all real numbers <math>k,</math> we have <math>a^{2k} + a^{-2k} + 2 = | + | Note that for all real numbers <math>k,</math> we have <math>a^{2k} + a^{-2k} + 2 = (a^{k} + a^{-k})^2,</math> from which <cmath>a^{2k} + a^{-2k} = (a^{k} + a^{-k})^2-2.</cmath> We apply this result twice to get the answer: |
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
− | a^4 + a^{-4} &= | + | a^4 + a^{-4} &= (a^2 + a^{-2})^2 - 2 \\ |
− | &= | + | &= [(a + a^{-1})^2 - 2]^2 - 2 \\ |
&= \boxed{\textbf{(D)}\ 194}. | &= \boxed{\textbf{(D)}\ 194}. | ||
\end{align*}</cmath> | \end{align*}</cmath> |
Revision as of 23:55, 7 October 2021
Contents
Problem
Suppose that the number satisfies the equation
. What is the value of
?
Solution 1 (Decreases the Powers)
Note that for all real numbers we have
from which
We apply this result twice to get the answer:
~Azjps (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
Solution 2 (Increases the Powers)
Squaring both sides of gives
from which
Squaring both sides of gives
from which
~Rbhale12 (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
Solution 3 (Detailed Explanation of Solution 2)
The detailed explanation of Solution 2 is as follows:
~MathFun1000 (Solution)
~MRENTHUSIASM (Minor Formatting)
Solution 4 (Binomial Theorem)
Squaring both sides of gives
from which
Applying the Binomial Theorem, we raise both sides of to the fourth power:
~MRENTHUSIASM
Solution 5 (Solves for a)
We multiply both sides of by
then rearrange:
We apply the Quadratic Formula to get
By Vieta's Formulas, note that the roots are reciprocals of each other. Therefore, both values of produce the same value of
Remarks about
- To find the fourth power of a sum/difference, we can first square that sum/difference, then square the result.
- When we expand the fourth powers and combine like terms, the irrational terms will cancel.
~Azjps (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
Solution 6 (Newton's Sums)
From the first sentence of Solution 4, we conclude that and
are the roots of
Let
By Newton's Sums, we have
~Albert1993 (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
Solution 7 (Answer Choices)
Note that We guess that
is an integer, so the answer must be
less than a perfect square. The only possibility is
~Thanosaops (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
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.