2010 AIME II Problems/Problem 5
Positive numbers , , and satisfy and . Find .
Using the properties of logarithms, by taking the log base 10 of both sides, and by using the fact that .
Through further simplification, we find that . It can be seen that there is enough information to use the formula , as we have both and , and we want to find .
After plugging in the values into the equation, we find that is equal to .
However, we want to find , so we take the square root of , or .
Let , and .
We have and . Since these two equations look a lot like Vieta's for a cubic, create the polynomial (leave the constant term as to make things easy). Dividing by yields .
Now we use the quadratic formula:
Since the question asks for (remember one of the values was the solution that we divided out in the beginning), we find:
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