2007 AMC 10A Problems/Problem 20
Contents
Problem
Suppose that the number satisfies the equation
. What is the value of
?
Solution 1 (Direct)
Notice 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
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 4
(similar to Solution 1)
We know that . We can square both sides to get
, so
. Squaring both sides again gives
, so
.
Solution 5
We let and
be roots of a certain quadratic. Specifically
. We use Newton's Sums given the coefficients to find
.
Solution 6
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 | |
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All AMC 10 Problems and Solutions |
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