# How do you combine (x^2)/(x^2 - 49) - x/(x-7)?

Apr 16, 2018

$- \frac{7 x}{\left(x - 7\right) \left(x + 7\right)}$

$\implies - \frac{7 x}{{x}^{2} - 49}$

#### Explanation:

First, you have to realize that the denominator of the first fraction is a difference between two squares:

If you have
$\left({a}^{2} - {b}^{2}\right)$

You can rewrite it as
$\left(a - b\right) \left(a + b\right)$

In this situation
$a = x$
$b = 7$

So we can rewrite the first fraction as
${x}^{2} / \left(\left(x - 7\right) \left(x + 7\right)\right)$

Now we have
${x}^{2} / \left(\left(x - 7\right) \left(x + 7\right)\right) - \frac{x}{x - 7}$

So in order to combine them, we need an LCD. In order to do so, we can multiply the second fraction by $\frac{x + 7}{x + 7}$. This works because it is the same thing as multiplying it by $1$, which wouldn't change it.

Now we have

${x}^{2} / \left(\left(x - 7\right) \left(x + 7\right)\right) - \left(\frac{x + 7}{x + 7}\right) \frac{x}{x - 7}$

$\implies {x}^{2} / \left(\left(x - 7\right) \left(x + 7\right)\right) - \frac{x \left(x + 7\right)}{\left(x - 7\right) \left(x + 7\right)}$

Now distribute the $x$ in the numerator of the second fraction

${x}^{2} / \left(\left(x - 7\right) \left(x + 7\right)\right) - \frac{{x}^{2} + 7 x}{\left(x - 7\right) \left(x + 7\right)}$

Now subtract the fractions

$\frac{{x}^{2} - {x}^{2} - 7 x}{\left(x - 7\right) \left(x + 7\right)}$

$\implies - \frac{7 x}{\left(x - 7\right) \left(x + 7\right)}$

$- \frac{7 x}{\left(x - 7\right) \left(x + 7\right)}$

or

$- \frac{7 x}{{x}^{2} - 49}$

Apr 16, 2018

$- \frac{7 x}{\left(x - 7\right) \left(x + 7\right)}$
or
$- \frac{7 x}{{x}^{2} - 49}$

#### Explanation:

$\frac{{x}^{2}}{{x}^{2} - 49} - \frac{x}{x - 7}$
Notice that ${x}^{2} - 49$ is a difference of squares (${x}^{2} - {7}^{2}$), so it can be factored to $\left(x + 7\right) \left(x - 7\right)$ Inputting this,
$\frac{{x}^{2}}{\left(x + 7\right) \left(x - 7\right)} - \frac{x}{x - 7}$
Now, multiply $\frac{x}{x - 7}$ by $\frac{x + 7}{x + 7}$ to get common denominators:
$\frac{{x}^{2}}{\left(x + 7\right) \left(x - 7\right)} - \frac{\left(x\right) \left(x + 7\right)}{\left(x - 7\right) \left(x + 7\right)}$
Since the denominators are now common, you can subtract the numerators and you get
$\frac{\left({x}^{2}\right) - \left(\left(x\right) \left(x + 7\right)\right)}{\left(x - 7\right) \left(x + 7\right)}$. This simplifies to
$- \frac{7 x}{\left(x - 7\right) \left(x + 7\right)}$
or
$- \frac{7 x}{{x}^{2} - 49}$