2005 AMC 10A Problems/Problem 10
Contents
Problem
There are two values of for which the equation
has only one solution for
. What is the sum of those values of
?
Solution 1
A quadratic equation has exactly distinct root if and only if the left-hand side is a perfect square. So we require
Two polynomials are equal only if their coefficients are equal, so we must have and
, which reduce to
and
respectively. By equating
-coefficients, it follows that
, so either
or
.
Accordingly, the desired sum is .
(Alternatively, we can observe that whatever the values of
are, they must lead to equations of the form
and
, corresponding to the perfect squares
. So the
choices of
, say
and
, must satisfy
and
, and adding these equations yields
.)
Solution 2
Since this quadratic must have a double root, its discriminant must be . Therefore we require
At this point, Vieta's formulae directly give the sum of the
possible values of
as
(Alternatively, we could use the quadratic formula to directly solve this equation for ; the precise value of the expression under the square root does not actually matter, as it will cancel out due to the
signs when added. This again gives the required sum as
See also
2005 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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All AMC 10 Problems and Solutions |
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