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MrRa [10]
2 years ago
11

Simplify :Stepwise answer! ​

Mathematics
1 answer:
Free_Kalibri [48]2 years ago
3 0

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

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