# How do you simplify \frac { 9x + 19} { x ^ { 2} + 2x - 3}?

Dec 4, 2017

It really doesn't simplify much. The bottom line becomes $\left(x + 3\right) \left(x - 1\right)$ and that's it.

#### Explanation:

You have a quadratic on the bottom, identifiable by the layout

${x}^{2} + x + \left(\text{NUMBER}\right)$

take your last number a.k.a the one without an unknown ' $- 3$ ' and multiply by your ${x}^{2}$ coefficient (how many of ${x}^{2}$ there are), in this case '$1$'. That gives you $- 3 \times 1 = - 3$

Then find the factors of this new value which add up to your $x$ coefficient, in this case $2$.

Factors of $- 3 = \left\{1 , - 3\right\} \mathmr{and} \left\{- 1 , 3\right\}$

$3$ and $- 1$ gives $\to 2$

Rewrite the original quadratic: ${x}^{2} + 2 x - 3$

as

${x}^{2} + 3 x - 1 x - 3 \to$ this came from the factors that worked.

remember that these two lines above are equal, but this second version makes it possible to simplify.

Take out what is common in the first two terms firstly, and then the second two terms.

$x \left(x + 3\right) - 1 \left(x + 3\right)$

we now have a common sum in both brackets, so we can actually rewrite this as:

$\left(x - 1\right) \left(x + 3\right)$ by taking what is outside each bracket and placing them together in their own set of brackets.

Lastly, don't forget that this is part of your original fraction.

The top line cannot simplify because $9 x$ and $19$ have nothing in common.

Viola!