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### Algebra: Common Denominators

```
Date: 5/14/96 at 15:43:56
From:  RDIANNE   GREEN
Subject: Algebra help needed, please :)

Hi. I am having problems working out the following two problems
because I don't know how to get all of them to have equal
denominators.

1.    2             a  +  1           4
-------   +    ----------   =    -----
a - 3          a^2 - 9          a + 3

2.   x            1
-----   =    -----
x + 2          x

```

```
Date: 5/14/96 at 21:12:59
From: Doctor Jodi
Subject: Re: Algebra help needed, please :)

Hi there! You may remember that one common denominator (though not
always the lowest) is found by multiplying the two (or more)
denominators together.

For example, for the fractions 2/7 and 1/3, you can multiply 7 * 3 to
get 21, one common denominator.

For your first problem, a^2 - 9 will be the lowest common denominator.
[I know this because of the "difference of squares" factoring rule.
Since a^2 - 9 is the difference of squares, we can factor it as
(a - 3)(a+3)]

So for the first problem, we will need to multiply fractions with the
denominator of a-3 by a+3, (which equals 1) and fractions with the
---
a+3

denominator of a+3 by a-3 (also equal to 1).  Does this make sense?
---
a-3

2       a + 3  =  2(a+3)    = 2a+6
--   * ------     ------      -----
a-3     a + 3     a^2 -9      a^2 -9

We can leave a+1     as it is for the moment.
---
a^2 -9

We will need to multiply

4          a-3                4 (a-3)         4a -12
-     *    ---     to get     -------    =    ------
a+3        a-3                 a^2 -9         a^2 -9

2a +6 +a+1  = 4a -12
-----------    ------
a^2 -9         a^2 -9

Does that look better?

You can now multiply through by a^2 -9 to remove the denominators if
you like and continue solving this like any other algebra problem.
Are you comfortable with the rest of the problem?  If not, write us
back and we'll give you a hand.

Here's just a clue for the second problem.  Again, if you want more
help, let us know.  For the second problem, the common denominator is
x(x+2) (aka x^2 + 2x)

Good luck and thanks for your question!
-Doctor Jodi,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Algebra

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