How do you combine 3/(x+2) - 2/(x^2+x-2) + 2/(x-1)?

May 25, 2015

You need to find the lowest common denominator between these fractions. To simplify things for us, we can factor the second fraction's denominator:

$\frac{- 1 \pm \sqrt{1 - 4 \left(1\right) \left(- 2\right)}}{2} = \frac{- 1 \pm 3}{2}$
${x}_{1} = 1$, thus being the factor $\left(x - 1\right) = 0$
${x}_{2} = - 2$, thus being the factor $\left(x + 2\right) = 0$

Rewriting everything, we have that the l.c.d. is $\left(x - 1\right) \left(x + 2\right)$.

$\frac{3}{x + 2} - \frac{2}{\left(x - 1\right) \left(x + 2\right)} + \frac{2}{x - 1}$

$\frac{3 \left(x - 1\right) - 2 + 2 \left(x + 2\right)}{\left(x - 1\right) \left(x + 2\right)}$=$\frac{3 x - 3 - 2 + 2 x + 4}{\left(x - 1\right) \left(x + 2\right)}$=$\frac{5 x - 1}{{x}^{2} + x - 2}$