If ( , ) are the roots of (x^2+b x+c=0, , ) are the roots of (x^2+b₁ x+c₁=0 ) and ( < < < ), then ( (c-c₁ )^2 < )

Step-by-step solution

According to given informations after drawing figure. For (x )-coordinate of point (P ), on subtracting given quadratic equations, we get ( aligned x^2+b₁ x+c₁ & =0 x^2+b x+c & =0 aligned ) ( aligned & (b₁-b ) x+ (c₁-c )=0 & x= ( c-c₁ b₁-b ) & aligned ) Now, with respect to quadratic expression ( gathered f(x)=x^2+b x+c f (x= c-c₁ b₁-b ) < 0 ( c-c₁ b₁-b )^2+b ( c-c₁ b₁-b )+c < 0 (c-c₁ )^2 < b (c₁-c ) (b₁-b )-c (b₁-b )^2 (c-c₁ )^2 < (b₁-b ) [b c₁-b c-c b₁+c b ] (c-c₁ )^2 < (b₁-b ) (b c₁-c b₁ ) gathered ) Hence, opti