# Factor and Simplify

#### PaperStSoap

##### New member
[(x2 - 4)(x2 + 3)1/2] - [(x2 - 4)2(x2 + 3)3/2]

I understand the first half which comes out to...

(x-2)(x+2)(x2 + 3)1/2

I have know idea how to start the second half.

#### CaptainBlack

##### Well-known member
[(x2 - 4)(x2 + 3)1/2] - [(x2 - 4)2(x2 + 3)3/2]

I understand the first half which comes out to...

(x-2)(x+2)(x2 + 3)1/2

I have know idea how to start the second half.

Look for the greatest common factor of the two terms, in this case that is $$x^2-4)(x^2+3)^{1/2}$$, so:
\begin{aligned}\left((x^2 - 4)(x^2 + 3)^{1/2}\right) - \left((x^2 - 4)^2(x^2 + 3)^{3/2}\right)&=(x^2-4)(x^2+3)^{1/2}\left(1-(x^2-4)(x^2+3) \right)\\ &=(4-x^2)(x^2+3)^{1/2}\left((x^2-4)(x^2+3)-1 \right) \end{aligned}

CB

#### soroban

##### Well-known member
Hello, PaperStSoap!

$$\text{Simplify: }\: (x^2-4)(x^2+3)^{\frac{1}{2}} - (x^2-4)^2(x^2+3)^{\frac{3}{2}}$$

Factor: .$(x^2-4)(x^2+3)^{\frac{1}{2}}\cdot\big[1 - (x^2-4)(x^2+3)\big]$

. . . . $=\;(x^2-4)(x^2+3)^{\frac{1}{2}}\cdot\big[1 - x^4 + x^2 + 12\big]$

. . . . $=\;(x^2-4)(x^2+3)^{\frac{1}{2}}\left(13 + x^2 - x^4\right)$