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

2 Answers
Jun 1, 2017

(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.

Jun 12, 2017

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