2013 AMC 10B Problems/Problem 19
Problem
The real numbers form an arithmetic sequence with
The quadratic
has exactly one root. What is this root?
Solution
It is given that has 1 real root, so the discriminant is zero, or
. Because a, b, c are in arithmetic progression,
, or
. We need to find the unique root, or
(discriminant is 0). From
, we have
. Ignoring the negatives, we have
. Fortunately, finding
is not very hard. Plug in
to
, we have
, or
, and dividing by
gives
, so
. But
, violating the assumption that
. Therefore,
. Plugging this in, we have
. But we need the negative of this, so the answer is
.