2005 AMC 10A Problems/Problem 10
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 .