Question

simplify.
1/x-2 +4/4-x^2
​

Answer 1

Given :  1/x-2 +4/4-x^2

To find :  Simplify

Solution:

1/(x-2) + 4/(4-x²)

=  1/(x-2) + 4/(2²-x²)

Taking -ve common from 2nd term

= 1/(x-2) - 4/(x² - 2²)

using a² - b² = (a + b)(a - b)

=  1/(x-2) - 4/(x + 2)(x - 2)

Taking  1/(x-2) common

= (1/(x - 2)) ( 1  - 4/(x + 2))

= (1/(x - 2)) ( (x + 2  - 4)/(x + 2))

= (1/(x - 2)) ( (x -2)/(x + 2))

Cancelling  x -2 from numerator and denominator

=  1/(x + 2)

1/(x-2) + 4/(4-x²) =  1/(x + 2)

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Answer 2

SOLUTION

TO SIMPLIFY

$\displaystyle \sf{ \frac{1}{x - 2} + \frac{4}{4 - {x}^{2} } }$

FORMULA TO BE IMPLEMENTED

We are aware of the formula

$\displaystyle \sf{ {a}^{2} - {b}^{2} = (a + b)(a - b) }$

EVALUATION

$\displaystyle \sf{ \frac{1}{x - 2} + \frac{4}{4 - {x}^{2} } }$

$\displaystyle \sf{ = \frac{1}{x - 2} - \frac{4}{{x}^{2} - 4} }$

$\displaystyle \sf{ = \frac{1}{x - 2} - \frac{4}{{x}^{2} - {2}^{2} } }$

$\displaystyle \sf{ = \frac{1}{x - 2} - \frac{4}{(x + 2)(x - 2) } }$

$\displaystyle \sf{ = \frac{(x + 2) - 4}{(x + 2)(x - 2) } }$

$\displaystyle \sf{ = \frac{(x + 2 - 4)}{(x + 2)(x - 2) } }$

$\displaystyle \sf{ = \frac{(x - 2)}{(x + 2)(x - 2) } }$

$\displaystyle \sf{ = \frac{1}{(x + 2)} }$

FINAL ANSWER

$\boxed{ \: \: \displaystyle \sf{ \frac{1}{x - 2} + \frac{4}{4 - {x}^{2} } } = \frac{1}{x + 2} \: \: }$

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