If the roots of the equation `a(b-c)x^(2) + b(c-a) x+c(a-b)=0` are equal, then prove that a, b, c are in H.P

If c, d are the roots of the equation (x-a) (x-b) - k=0 , then prove that a, b are the roots of the equation (x-c) (x-d) + k= 0

If (1)/(b-a)+ (1)/(b-c) = (1)/(a) +(1)/(c ) then prove that a, b,c are in H.P

If a (1/b+1/c), b(1/c+1/a), c(1/a+1/b) are in A.P. Prove that a, b, c are in A.P.

If the roots of the equation (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 are equal, then a^(2)+b^(2)+c^(2) is equal to

If the roots of the equation x ^(2) + 2 bx + c =0 are alpha and beta, then b ^(2) - c is equal to

If the roots of the equation x^(2) - bx + c = 0 are two consecutive integers, then b^(2) - 4c is