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

2 Answers
Apr 16, 2018

-(7x)/((x-7)(x+7))

=>-(7x)/(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
(a^2-b^2)

You can rewrite it as
(a-b)(a+b)

In this situation
a=x
b=7

So we can rewrite the first fraction as
x^2/((x-7)(x+7))

Now we have
x^2/((x-7)(x+7))-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 (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/((x-7)(x+7))-((x+7)/(x+7))x/(x-7)

=>x^2/((x-7)(x+7))-(x(x+7))/((x-7)(x+7))

Now distribute the x in the numerator of the second fraction

x^2/((x-7)(x+7))-(x^2+7x)/((x-7)(x+7))

Now subtract the fractions

(x^2-x^2-7x)/((x-7)(x+7))

=>-(7x)/((x-7)(x+7))

Answer

-(7x)/((x-7)(x+7))

or

-(7x)/(x^2-49)

Apr 16, 2018

-(7x)/((x-7)(x+7))
or
-(7x)/(x^2-49)

Explanation:

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