# How do you solve 5/(x^2+x-6) = 2 + (x-3)/(x-2) and find any extraneous solutions?

Jul 11, 2017

color(blue)(x=-1/3+sqrt(79)/3or x=-1/3-sqrt(79)/3

#### Explanation:

$\frac{5}{{x}^{2} + x - 6} = 2 + \frac{x - 3}{x - 2}$

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

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

multiply both sides by $\left(x + 3\right) \left(x - 2\right)$

$\therefore 5 = 2 \left(x + 3\right) \left(x - 2\right) + \left(x - 3\right) \left(x + 3\right)$

$\therefore 5 = 2 {x}^{2} + 2 x - 12 + {x}^{2} - 9$

$\therefore 5 + 12 + 9 = 3 {x}^{2} + 2 x$

$\therefore 3 {x}^{2} + 2 x = 26$

$\therefore 3 {x}^{2} + 2 x - 26 = 0$

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$a = 3 , b = 2 , c = - 26$

$\therefore x = \frac{- \left(2\right) \pm \sqrt{{\left(2\right)}^{2} - 4 \left(3\right) \left(- 26\right)}}{2 \left(3\right)}$

$\therefore x = \frac{- 2 \pm \sqrt{316}}{6}$

$\therefore \frac{- 2 \pm \sqrt{79 \cdot 2 \cdot 2}}{6}$

$\therefore \frac{- 2 \pm 2 \sqrt{79}}{6}$

$\therefore x = \frac{- 2 + 2 \sqrt{79}}{6} \mathmr{and} x = \frac{- 2 - 2 \sqrt{79}}{6}$

:.color(blue)(x=-1/3+(sqrt(79))/3 or x=-1/3-sqrt(79)/3