How do you simplify (2x + 3)/( x + 3) + x/ (x - 2)?

Oct 28, 2017

It is a little strange:

$\left(\frac{1}{3}\right) \frac{\left(3 x + \left(1 + \sqrt{19}\right)\right) \left(3 x + \left(1 - \sqrt{19}\right)\right)}{\left(x + 3\right) \left(x - 2\right)}$

Explanation:

$\frac{2 x + 3}{x + 3} + \frac{x}{x - 2}$

$= \frac{x - 2}{x - 2} \frac{2 x + 3}{x + 3} + \frac{x + 3}{x + 3} \frac{x}{x - 2}$

$= \frac{2 {x}^{2} + 3 x - 4 x - 6 + {x}^{2} + 3 x}{\left(x + 3\right) \left(x - 2\right)}$

$= \frac{3 {x}^{2} + 2 x - 6}{\left(x + 3\right) \left(x - 2\right)}$

Now we must find the roots of:

the denominator:

$x = \frac{- 2 \pm \sqrt{{2}^{2} - 4 \cdot 3 \cdot \left(- 6\right)}}{2 \cdot 3}$

$x = \frac{- 2 \pm \sqrt{76}}{6}$

$x = \frac{- 1 \pm \sqrt{19}}{3}$

So the polynomium simplifies to:

$3 \frac{\left(x + \frac{1 + \sqrt{19}}{3}\right) \left(x + \frac{1 - \sqrt{19}}{3}\right)}{\left(x + 3\right) \left(x - 2\right)}$

$\left(\frac{1}{3}\right) \frac{\left(3 x + \left(1 + \sqrt{19}\right)\right) \left(3 x + \left(1 - \sqrt{19}\right)\right)}{\left(x + 3\right) \left(x - 2\right)}$