Difference between revisions of "2007 AMC 10A Problems/Problem 20"
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Revision as of 11:51, 4 July 2013
Problem
Suppose that the number satisfies the equation
. What is the value of
?
Solution
Solution 1
Notice that . Thus
.
Solution 2
. 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
We know that . We can square both sides to get
, so
. Squaring both sides again gives
, so
. Q. E. D.
Solution 4
We let and
be roots of a certain quadratic. Specifically
. We use Newton's Sums given the coefficients to find
.
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.