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2021 Fall AMC 10A Problems/Problem 20

Revision as of 19:03, 22 November 2021 by MRENTHUSIASM (talk | contribs)

Problem

How many ordered pairs of positive integers $(b,c)$ exist where both $x^2+bx+c=0$ and $x^2+cx+b=0$ do not have distinct, real solutions?

$\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 12 \qquad$

Solution

A quadratic equation has no real solutions if and only if the discriminant is nonpositive. Therefore:

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