# How do you simplify (15x^2-8x-18)/(-20x^2+14x+12)?

Sep 30, 2016

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#### Explanation:

The only way to simplify a rational polynomial expression is to group something, usualy by finding the solutions, or any other factoring technique.

For example, we can factor a $2$ in the denominator, so that it becomes

$2 \left(- 10 {x}^{2} + 7 x + 6\right)$.

As for the solutions, the denominator looks pretty ugly: we have

$15 {x}^{2} - 8 x - 18 = 0 \setminus \iff x = \frac{4}{15} \setminus \pm \frac{\sqrt{286}}{15}$

The denominator is much easier:

$- 10 {x}^{2} + 7 x + 6 = 0 \setminus \iff x = - \frac{1}{2}$ or $\frac{6}{5}$

This means that we can write our fraction as

$\frac{\left(x - \frac{4}{15} - \frac{\sqrt{286}}{15}\right) \left(x - \frac{4}{15} + \frac{\sqrt{286}}{15}\right)}{2 \left(- 2 x + 1\right) \left(5 x - 6\right)}$

And there's nothing we can simplify.

As you can see here WolframAlpha can only group something in the numerator or denominator, but can't cancel anything that is present in both.