What is (sqrt(1-x^2)/2) - (x^2-x/sqrt(1-x^2))?

Expert Answers

Simplify `sqrt(1-x^2)/2 - (x^2-x)/sqrt(1-x^2)`

Generally you want to write as a single term if possible, with no radicals in the denominator.

One approach is to rationalize the second term and add the fractions:

`sqrt(1-x^2)/2-(x^2-x)/sqrt(1-x^2)=sqrt(1-x^2)/2-(x^2-x)/sqrt(1-x^2)sqrt(1-x^2)/sqrt(1-x^2)`

`=sqrt(1-x^2)/2-((x^2-x)sqrt(1-x^2))/(1-x^2)`  The common denominator is `2(1-x^2)` :

`=(sqrt(1-x^2)(1-x^2)-2(x^2-x)sqrt(1-x^2))/(2(1-x^2))`

`=(sqrt(1-x^2)(1-x^2-2x^2+2x))/(2(1+x)(1-x))`

`=(sqrt(1-x^2)(1+3x)(1-x))/(2(1+x)(1-x))`

`=((3x+1)sqrt(1-x^2))/(2(1+x))` which does not simplify further.

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Simplify `sqrt(1-x^2)/2 - (x^2-x)/sqrt(1-x^2)`

Generally you want to write as a single term if possible, with no radicals in the denominator.

One approach is to rationalize the second term and add the fractions:

`sqrt(1-x^2)/2-(x^2-x)/sqrt(1-x^2)=sqrt(1-x^2)/2-(x^2-x)/sqrt(1-x^2)sqrt(1-x^2)/sqrt(1-x^2)`

`=sqrt(1-x^2)/2-((x^2-x)sqrt(1-x^2))/(1-x^2)`  The common denominator is `2(1-x^2)` :

`=(sqrt(1-x^2)(1-x^2)-2(x^2-x)sqrt(1-x^2))/(2(1-x^2))`

`=(sqrt(1-x^2)(1-x^2-2x^2+2x))/(2(1+x)(1-x))`

`=(sqrt(1-x^2)(1+3x)(1-x))/(2(1+x)(1-x))`

`=((3x+1)sqrt(1-x^2))/(2(1+x))` which does not simplify further.

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