Please, HOW WOULD YOU SOLVE THIS?

Question

I need to add these two fractions!

-3/(5 - a) and 5/(a^2 - 25)

I think it has to do with factoring (a^2 - 25)....

Answer 1

You are correct. You will want to factor the denominator a² - 25. This is a difference of squares which would become:
(a - 5)(a + 5)

Now you almost have (a - 5) in the denominator of the first fraction, just the numbers are reversed...

For the first fraction, transfer the negative to the denominator; that will switch the terms:
. -3
--------
(5 - a)

... 3
----------
-(5 - a)

.. 3
--------
-5 + a

.. 3
--------
(a - 5)

Now to combine the fractions you need the common denominator (which you already figured out will be (a - 5)(a + 5). So multiply the first fraction by (a+5) in numerator and denominator. That will get you the following:
.. 3(a + 5) ............. 5
----------------- + -----------------
(a - 5)(a + 5) .. (a - 5)(a + 5)

Add the numerators, put it over the common denominator:
3(a + 5) + 5
-----------------
(a - 5)(a + 5)

Distribute the 3 through the parentheses:
3a + 15 + 5
-----------------
(a - 5)(a + 5)

Simplify by addition:
... 3a + 20
-----------------
(a - 5)(a + 5)

I would leave the denominator factored, but you could put it back to the other form if you want. Either answer should be acceptable.

Answer 2

Factoring would be the easiest way:
a^2 - 25 = (a - 5) * (a + 5) = -1 * (5 - a) * (5 + a)

You can therefore get a common denominator in both addends, which is what you need to do to add them, by multiplying the top and bottom of the first fraction by
-1 * (5 + a)

The top becomes 15 + 3a, so adding the two numerators we get
3a + 15 + 5 = 3a + 20

and the answer is
(3a + 20) / (a^2 - 25)

Answer 3

The same way you add fractions in elementary school -- find a common denominator. Now, in this case, a² - 25 is a multiple of both a² - 25 and 5 - a, since (5-a)*(-5-a) = a²-25. So multiply both the numerator and denominator by -5-a to obtain:

-3(-5-a)/(a² - 25) + 5/(a² - 25)

Now add and simplify:

(-3(-5-a) + 5)/(a² - 25)
(3a + 15 + 5)/(a² - 25)
(3a+20)/(a² - 25)

And we are done.

Answer 4

yes, factor the denominator to find the LCD

-3/(5 -a) = 3/(a - 5) multiplying top and bottom by -1
5/(a^2 - 25) = 5/(a+5)(a-5)

LCD = (a - 5)(a + 5) so you need to multiply the first fraction by (a+5)

3(a+5)/(a+5)(a-5) + 5/(a+5)(a - 5)
= 3a + 15/ (a + 5)(a - 5) + 5/(a+5)(a - 5)
= (3a + 20)/(a+5)(a -5)

Answer 5

No, in order to add them they must have a common denominator. Multiply the first by (a^2-25)/(a^2-25) and the second by (5-a)/(5-a). Once you do that combine the like terms in the numerator and you'll have your answer.

Answer 6

You think wrong. Multiply by the reciprocal.