Simplifying [(x2 - 4)(x2 + 3)1/2] - [(x2 - 4)2(x2 + 3)3/2]

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The discussion focuses on simplifying the expression \([(x^2 - 4)(x^2 + 3)^{1/2}] - [(x^2 - 4)^2(x^2 + 3)^{3/2}]\). The first half simplifies to \((x-2)(x+2)(x^2 + 3)^{1/2}\). The second half is addressed by factoring out the greatest common factor, resulting in \((x^2 - 4)(x^2 + 3)^{1/2}\left(1 - (x^2 - 4)(x^2 + 3)\right)\), which further simplifies to \((x^2 - 4)(x^2 + 3)^{1/2}(13 + x^2 - x^4)\).

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[(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.

Thanks in advance.
 
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PaperStSoap said:
[(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.

Thanks in advance.

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
 
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] $

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

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

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