Difference between revisions of "2007 AMC 10A Problems/Problem 20"
MRENTHUSIASM (talk | contribs) m (→Solution 7 (Detailed Version of Solution 1)) |
MRENTHUSIASM (talk | contribs) m (→Solution 7 (Detailed Version of Solution 1): Used negative exponents instead of fractions, so the solution is consistent with the problem and the other solutions.) |
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The algebra is as follows: | The algebra is as follows: | ||
<cmath>\begin{alignat*}{8} | <cmath>\begin{alignat*}{8} | ||
− | a+ | + | a+a^{-1}&=4 \\ |
− | \left(a+ | + | \left(a+a^{-1}\right)^2&=4^2 \\ |
− | a^2+ | + | a^2+2aa^{-1}+a^{-2}&=16 \\ |
− | a^2+ | + | a^2+a^{-2}&=16-2&&=14 \\ |
− | \left(a^2+ | + | \left(a^2+a^{-2}\right)^2&=14^2 \\ |
− | + | a^4+2a^2a^{-2}+a^{-4}&=196 \\ | |
− | a^4+ | + | a^4+a^{-4}&=196-2&&=\boxed{\text{(D)}\ 194}. \\ |
\end{alignat*}</cmath> | \end{alignat*}</cmath> | ||
~MathFun1000 (Solution) | ~MathFun1000 (Solution) |
Revision as of 22:59, 6 July 2021
Contents
Problem
Suppose that the number satisfies the equation
. What is the value of
?
Solution 1 (Increases the Powers)
Squaring both sides of gives
from which
Squaring both sides of gives
from which
~Rbhale12 (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
Solution 2 (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 3 (Binomial Theorem)
Squaring both sides of gives
from which
Applying the Binomial Theorem, we raise both sides of to the fourth power:
~MRENTHUSIASM
Solution 4 (Solves for a)
We multiply both sides of by
then rearrange:
We apply the Quadratic Formula to get
Note that the roots are reciprocals of each other. Therefore, choosing either value for gives the same value for
Remarks in
- 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 5 (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 6 (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)
Solution 7 (Detailed Version of Solution 1)
The algebra is as follows:
~MathFun1000 (Solution)
~MRENTHUSIASM (Minor Formatting)
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.