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Algebra Difficulty 4.6 AIME Find the answer

Example 1. Solve the equation: 3xx2−1+x2−13x=−52\frac{3 x}{x^{2}-1}+\frac{x^{2}-1}{3 x}=-\frac{5}{2}.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solve the original equation ⟺ Proposition 2 3xx2−1=2\stackrel{\text { Proposition } 2 \text { }}{\Longleftrightarrow} \frac{3 x}{x^{2}-1}=2 or 3xx2−1=\frac{3 x}{x^{2}-1}= 12\frac{1}{2}, solving it yields x1=2,x2=−12,x3=3+10x_{1}=2, x_{2}=-\frac{1}{2}, x_{3}=3+\sqrt{10},

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