Help with adding rational expressions?

Question

2/ x^2-9 + 3/ x+3. Ok i know the first step is to factor, which i have 2/(x+3)(x-3) + x+3, now would my Least common denominator be (x-3) ? Or will it be (x+3)(x-3) please explain:) Thanks so much.

Answer 1

Least common denominator is (x-3)(x+3).

2/(x-3)(x+3) + 3*(x-3)/(x-3)(x+3)

(2+3x-9)/(x-3)(x+3)

(3x-7)/(x-3)(x+3).

Answer 2

Look at your first term. Are you able to cancel anything? No, because there are no common factors in the numerator and denominator. So your common denominator is at least the two factors:(x+3)(x-3).

Now look at your second term. You can see that you need an (x-3) in the denominator. {I assume it is just a typo that you dropped "3/" in your factored expression.}

The "least common denominator" stands for the ENTIRE common denominator, so it would include both factors needed in the denominator.


However, you are correct to observe that you only need to multiply by (x-3) to get the correct result.

No real harm would be done if you hadn't seen the common factor of (x+3) in your two terms. You would just be doing unneeded extra work.


Mathematicians are a LAZY bunch, and HATE extra work. (Which is really why they come up with ideas like the least common denominator). It seems like a lot of work, and unnecessarily complicated, ANY common denominator works, but the LEAST one is not only the smallest, it is also the one with the least work.

I took all the math I could in high school, because math teachers rarely made me write papers.

You might confirm your final answer by checking it.
One way is to pick some values to plug into your original expression, and your final expression. Avoid anything that would make any denominator go to zero. I would pick 0, a small positive number, and a small negative number and substitute for x.


Did I mention, that I am not only lazy, but also sloppy at math? ;)

Answer 3

(2/(x^2 - 9)) + (3/(x + 3))
(2/((x - 3)(x + 3))) + (3/(x + 3))
Multiply everything by (x - 3)(x + 3)
(2 + 3(x - 3))/((x - 3)(x + 3))
(2 + 3x - 9)/((x - 3)(x + 3))
(3x - 7)/((x - 3)(x + 3))
(3x - 7)/(x^2 - 9)

LCD is (x + 3)(x - 3)

The reason is because its the least value that both would have in common. If you only say (x - 3), which is the least common factor that they have in common, then you are only including one factor of x^2 - 9 and not both factors.

Answer 4

assuming that ur original eguation was

2 / ( x ^ 2 - 9 ) + 3 / ( x + 3 ) factor for

2 / [ ( x + 3 ) ( x - 3 ) ] + 3 / ( x + 3 ) common denominator would be

( x + 3 ) ( x - 3 ) so u would get

2 / [ ( x + 3 ) ( x - 3 ) ] + [ 3 ( x - 3 ) ] / [ ( x + 3 ) ( x - 3 ) ] finally getting

[ 2 + 3 x - 9] / [ ( x + 3 ) ( x - 3 ) ] or

[ 3 x - 7 ] / [ ( x + 3 ) ( x - 3 ) ]