2011 AMC 10A Problems/Problem 16
Contents
Problem 16
Which of the following is equal to ?
Solution 1 (Fast)
We find the answer by squaring, then square rooting the expression.
Solution 2 (FASTER!)
We can change the insides of the square root into a perfect square and then simplify.
Solution 3 (FASTEST!!)
Square roots remind us of squares. So lets try to make . Doing a little experimentation we find that Similarly since we know that
We want to find . Using what we found above we know This is nothing but .
~coolmath_2018
Note: This is basically just Solution 2 except you "do a little experimentation"
Solution 4 (No Words)
~JH. L
Solution 5 (Bash / Int. Alg.)
First, factor out of the whole expression: Let We square both sides, leaving us with We equate coefficients, and we see that and We can see a pair of values that works easily, or A quick sign check tells us that the valid solution is Similarly, for we get When we add these two, we get and multiplying by we get
We can quickly check our answer by estimating and : becomes This should be our approximate answer. We got - the same thing. ~Technodoggo
Video Solution
~savannahsolver
See Also
2011 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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All AMC 10 Problems and Solutions |
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