2007 AMC 10A Problems/Problem 20
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 | |
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All AMC 10 Problems and Solutions |
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