Track / Stage 4 / 154 of 340 #414 of 1964

Problem 414

AMC 12 late, AIME early

Algebra Difficulty 4.8 Find the answer

$1 . m$ is what integer when the equation
$\left(m^{2}-1\right) x^{2}-6(3 m-1) x+72=0$

has two distinct positive integer roots?

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

(Tip: $m^{2}-1 \neq 0, m \neq \pm 1$ . Since $\Delta=36(m-3)^{2}>$ 0, hence $m \neq 3$ . Using the quadratic formula, we get $x_{1}=\frac{6}{m-1}, x_{2}=$ $\frac{12}{m+1}$ . Therefore, $(m-1)|6,(m+1)| 12$ . So $m=2$ , at this point, $x_{1}=6, x_{2}=4$ .)

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.