Question

Simplify :
Stepwise answer! ​

Answer 1

Answer:

$\boxed{ \bold{ \huge{ \boxed{ \sf{ \frac{14 {x}^{2} - 96}{ {x}^{4} - 13 {x}^{2} + 36}}}}}}$

Step-by-step explanation:

$\sf{ \frac{2}{x - 2} + \frac{1}{x - 3} - \frac{2}{ x + 2} - \frac{1}{ x + 3}}$

⇒$\sf{ \frac{2(x - 3)(x + 3)(x + 2) + (x - 2)(x + 2)(x + 3) - 2(x - 2)(x - 3)(x + 3) - (x - 2)(x + 2)(x - 3)}{(x - 2)(x - 3)(x + 2)(x + 3)}}$

Use the formula : a² - b² = ( a + b ) ( a - b )

⇒$\sf{ \frac{2( {x}^{2} - 9)(x + 2) + ( {x}^{2} - 4)(x + 3) - (2x - 4)( {x}^{2} - 9) - ( {x}^{2} - 4)(x - 3) }{( {x}^{2} - 4)( {x}^{2} - 9)}}$

Distribute 2 through the parentheses

⇒$\sf{ \frac{(2 {x}^{2} - 18)(x + 2) + ( {x}^{2} - 4)(x + 3) - (2x - 4)( {x}^{2} - 9) - ( {x}^{2} - 4)(x - 3)}{( {x}^{2} - 4)( {x}^{2} - 9) } }$

Multiply the algebraic expressions

⇒$\sf{ \frac{2 {x}^{3} + 4 {x}^{2} - 18x - 36 + {x}^{3} + 3 {x}^{2} - 4x - 12 - (2 {x}^{3} - 18x - 4 {x}^{2} + 36) - ( {x}^{3} - 3 {x}^{2} - 4x + 12) } {( {x}^{2} - 4)( {x}^{2} - 9)} }$

When there is a ( - ) in front of an expression, change the sign of each term in the expression

⇒$\sf{ \frac{2 {x}^{3} + 4 {x}^{2} - 18x - 36 + {x}^{3} + 3 {x}^{2} - 4x - 12 - 2 {x}^{3} + 18x + 4 {x}^{2} - 36 - {x}^{3} + 3 {x}^{2} + 4x - 12}{( {x}^{2} - 4)( {x}^{2} - 9) } }$

Since two opposites add up to zero, it would be better to remove them from the expression

⇒$\sf{ \frac{4 {x}^{2} - 36 + 3 {x}^{2} - 12 + 4 {x}^{2} - 36 + 3 {x}^{2} - 12}{( {x}^{2} - 4)( {x}^{2} - 9)} }$

collect like terms and simplify

⇒$\sf{ \frac{14 {x}^{2} - 48 - 36 - 12}{( {x}^{2} - 4)( {x}^{2} - 9) } }$

⇒$\sf{ \frac{14 {x}^{2} - 84 - 12}{( {x}^{2} - 4)( {x}^{2} - 9) } }$

⇒$\sf{ \frac{14 {x}^{2} - 96}{ ({x}^{2} - 4)( {x}^{2} - 9)} }$

Multiply : ( x² - 4 ) and ( x² - 9 )

⇒$\sf{ \frac{14 {x}^{2} - 96}{ {x}^{4} - 13 {x}^{2} + 36}}$

Hope I helped!

Best regards! :D