Let and be the distinct roots of a x^2+b x+c=0 , then _ x 1- (a x^2+b x+c ) (x- )^2 is equal to

Step-by-step solution

The quadratic equation ax^2 + bx + c = 0 has roots and . Thus, ax^2 + bx + c = a(x - )(x - ) . The limit is given by L = _ x 1 - (a(x - )(x - )) (x - )^2 . Using the identity 1 - = 2 ^2( /2) , we have L = _ x 2 ^2( a(x - )(x - ) 2 ) (x - )^2 . Using the standard limit _ u 0 u u = 1 , we multiply and divide by the square of the argument of the sine function: L = _ x 2 [ ( a(x - )(x - ) 2 ) a(x - )(x - ) 2 ]^2 a^2(x - )^2(x - )^2 4(x - )^2 . As x , the term in the square brackets approaches 1. The expression simplifi