Simplify the expression [tex]\frac{x^2 - x - 12}{2x^2 + 2x - 12}[/tex].

Please provide a clear explanation. Thank you.

For x^2 - x - 12, the discriminant is ( - 1 ) ^2 - 4 * 1 * ( -12 ) = 49 => sqrt(49) = 7 => x1 = ( 1 + 7 ) / 2 = 4 and x2 = ( 1 - 7 ) / 2 => x2 = - 3 => x^2 - x - 12 = ( x - 4 )( x + 3 ) ;
In the same way, 2x^2+2x-12 = 2( x - 2 )( x + 3 );Then , (X^2-x-12) / (2x^2+2x-12) ==[ ( x - 4 )( x + 3 )] / [2( x - 2 )( x + 3 )] => ( x - 4 ) / ( 2x - 4 ), where x isn' t 2 or - 3.

Answered by crisforp | 2024-06-10

To simplify the expression 2 x 2 + 2 x − 12 x 2 − x − 12 ​ , we factor the numerator as ( x − 4 ) ( x + 3 ) and the denominator as 2 ( x + 3 ) ( x − 2 ) . After canceling out the common factor ( x + 3 ) , the simplified expression is 2 ( x − 2 ) x − 4 ​ . This is valid for all x except for x = − 3 and x = 2 .
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Answered by crisforp | 2024-12-26