2021 Fall AMC 12A Problems/Problem 17

Revision as of 19:54, 23 November 2021 by Ehuang0531 (talk | contribs) (Created page with "==Problem== For how many ordered pairs <math>(b,c)</math> of positive integers does neither <math>x^2+bx+c=0</math> nor <math>x^2+cx+b=0</math> have two distinct real solution...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions?

$\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16 \qquad$

Solution

If a quadratic equation does not have two distinct real solutions, then its discriminant must be $\le0$. So, $b^2-4c\le0$ and $c^2-4b\le0$.