Question

How do you simplify `(4x^2)/(4x^2-9)`?

Answer 1

`(4x^2)/((2x+3)(2x-3)`

This expression does not simplify.

Explanation:

`(4x^2)/(4x^2-9)`

Do not be tempted to cancel the `4x^2` in the numerator and denominator. The fact that there are 2 terms in the denominator means you cannot cancel.

Perhaps factorising the denominator will help?

`(4x^2)/((2x+3)(2x-3)`

This expression does not simplify.

Answer 2

Another way of writing this which might be considered "simple" is to decompose this into partial fractions.

This gives the answer `3/(2(2x-3))-3/(2(2x+3))`.

Explanation:

(Note that for all intents and purposes, `(4x^2)/((2x-3)(2x+3)` is perfectly fine. This method just shows another way to simplify a rational function.)

The separation through partial fractions will look like this:

`(4x^2)/((2x+3)(2x-3)) = A/(2x+3) + B/(2x-3)`

`4x^2 = A(2x-3) + B(2x+3)`

Let `x=3/2`:

`4(3/2)^2 = A(2(3/2)-3) + B(2(3/2)+3)`

`4(9/4) = 0A+6B`

`9 = 6B`

`3/2 = B`

Let `x=-3/2`

`4(-3/2)^2 = A(2(-3/2)-3)+B(2(-3/2)+3)`

`4(9/4) = -6A + 0B`

`9 = -6A`

`-3/2 = A`

Therefore:

`(4x^2)/((2x+3)(2x-3)) = (-3/2)/(2x+3) + (3/2)/(2x-3)`

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

Final Answer