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Revision as of 21:46, 3 August 2006
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
.