Equation with algebraic fractions

Statement

Solve: $\frac{x+3}{x^4-x^2} - \frac{1}{x^2+x} = \frac{2}{x^2} - \frac{3}{x^2-x}$

Solution

E.C.: $x\neq 0$, $x\neq 1$, $x\neq -1$.

Factorise all the denominators: $\frac{x+3}{x^2(x+1)(x-1)} - \frac{1}{x(x+1)} = \frac{2}{x^2} - \frac{3}{x(x-1)}$ The least common multiple of the denominators is $\text{mcm} = x^2(x+1)(x-1).$ Reducing to the same denominator and equating the numerators gives: $(x+3) - x(x-1) = 2(x+1)(x-1) - 3x(x+1)$ Expand and solve the resulting equation, discarding any solutions that are not compatible with the existence conditions.

Links

Topics: Algebraic fractions
Concepts: Existence conditions · Fractional equation · Algebraic fraction · Least common multiple (LCM)
Skills: Solving equations · Factorising · Simplifying
Exercise type: Solving an equation