Difference between revisions of "2013 AMC 10B Problems/Problem 19"
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− | The problem says | + | The problem says <math>a\ge b\ge c</math>, but you assumed that <math>a\le b\le c</math>. I know many people who made this mistake. |
Revision as of 18:58, 26 February 2013
Problem
The real numbers form an arithmetic sequence with
The quadratic
has exactly one root. What is this root?
I didn't really understand this problem...
If there is one root, the discriminant is 0. Therefore, . Let's define
and
. If
, by substitution,
and distributing,
. Thus, through simple arithmetic,
and
. Then, the root, by quadratic formula, is
. Since the discriminant is assumed as zero, the root will be
, or
or
. This is equal to
or
. Multiplying by the conjugate gives
.
This would mean the answer would be , but the answer key says
. Any criticism would be much welcomed.
The problem says , but you assumed that
. I know many people who made this mistake.