# How do you simplify \frac { 4} { 3x - 6} - \frac { 5} { 4x + 4} + \frac { 2} { 5x + 10}?

Sep 20, 2017

$\frac{29 {x}^{2} + 216 x + 412}{\left(4 x + 4\right) \left(5 x + 10\right) \left(3 x - 6\right)}$

#### Explanation:

You first have to make every denominator the same, so multiply all the fractions by another fraction that equals one.

$= \frac{4}{3 x - 6} \cdot \frac{\left(4 x + 4\right) \left(5 x + 10\right)}{\left(4 x + 4\right) \left(5 x + 10\right)} - \frac{5}{4 x + 4} \frac{\left(3 x - 6\right) \left(5 x + 10\right)}{\left(3 x - 6\right) \left(5 x + 10\right)} + \frac{2}{5 x + 10} \frac{\left(3 x - 6\right) \left(4 x + 4\right)}{\left(3 x - 6\right) \left(4 x + 4\right)}$

$= \frac{4 \left(4 x + 4\right) \left(5 x + 10\right) - 5 \left(5 x + 10\right) \left(3 x - 6\right) + 2 \left(3 x - 6\right) \left(4 x + 4\right)}{\left(4 x + 4\right) \left(5 x + 10\right) \left(3 x - 6\right)}$

You get a really large fraction, but from here you just expand simplify. You don't have to expand the denominator because nothing factors easily yet.

$= \frac{4 \left(20 {x}^{2} + 60 x + 40\right) - 5 \left(15 {x}^{2} - 60\right) + 2 \left(12 {x}^{2} - 12 x - 24\right)}{\left(4 x + 4\right) \left(5 x + 10\right) \left(3 x - 6\right)}$
=(80x^2+240x+160-75x^2+300+24x^2-24x-48)/((4x+4)(5x+10)(3x-6)
$= \frac{29 {x}^{2} + 216 x + 412}{\left(4 x + 4\right) \left(5 x + 10\right) \left(3 x - 6\right)}$

Using the quadratic formula, we can tell what the roots are for the quadratic in the numerator.

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a} = \frac{- 216 \pm \sqrt{{216}^{2} - 4 \left(29\right) \left(412\right)}}{2 \left(29\right)} = \frac{- 216 \pm \sqrt{46656 - 47792}}{58} = \frac{- 216 \pm \sqrt{- 1136}}{58}$

Since taking the square root of a negative number gives us imaginary answers, we can conclude that the quadratic cannot factor further, and this is the final, simplified fraction:
$\frac{29 {x}^{2} + 216 x + 412}{\left(4 x + 4\right) \left(5 x + 10\right) \left(3 x - 6\right)}$