2005 AMC 10A Problems/Problem 10
From AoPSWiki
Problem
There are two values of 
for which the equation 
has only one solution for 
. What is the sum of those values of ?
Solution
A quadratic equation has exactly one root if and only if it is a perfect square. So set
Two polynomials are equal only if their coefficients are equal, so we must have
or 
.
So the desired sum is 
Alternatively, note that whatever the two values of are, they must lead to equations of the form 
and 
. So the two choices of must make 
and 
so 
and 
.
