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

Jun 1, 2017

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

This expression does not simplify.

#### Explanation:

$\frac{4 {x}^{2}}{4 {x}^{2} - 9}$

Do not be tempted to cancel the $4 {x}^{2}$ 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 $\frac{3}{2 \left(2 x - 3\right)} - \frac{3}{2 \left(2 x + 3\right)}$.

#### 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:

$\frac{4 {x}^{2}}{\left(2 x + 3\right) \left(2 x - 3\right)} = \frac{A}{2 x + 3} + \frac{B}{2 x - 3}$

$4 {x}^{2} = A \left(2 x - 3\right) + B \left(2 x + 3\right)$

Let $x = \frac{3}{2}$:

$4 {\left(\frac{3}{2}\right)}^{2} = A \left(2 \left(\frac{3}{2}\right) - 3\right) + B \left(2 \left(\frac{3}{2}\right) + 3\right)$

$4 \left(\frac{9}{4}\right) = 0 A + 6 B$

$9 = 6 B$

$\frac{3}{2} = B$

Let $x = - \frac{3}{2}$

$4 {\left(- \frac{3}{2}\right)}^{2} = A \left(2 \left(- \frac{3}{2}\right) - 3\right) + B \left(2 \left(- \frac{3}{2}\right) + 3\right)$

$4 \left(\frac{9}{4}\right) = - 6 A + 0 B$

$9 = - 6 A$

$- \frac{3}{2} = A$

Therefore:

$\frac{4 {x}^{2}}{\left(2 x + 3\right) \left(2 x - 3\right)} = \frac{- \frac{3}{2}}{2 x + 3} + \frac{\frac{3}{2}}{2 x - 3}$

$\frac{3}{2 \left(2 x - 3\right)} - \frac{3}{2 \left(2 x + 3\right)}$