If roots of equation `x^2+2c x+a b=0` are real and unequal, then prove that the roots of `x^2-2(a+b)x+a^2+b^2+2c^2=0` will be imaginary.

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

If the roots of the equation (c^(2)-ab)x^(3)-2(a^(2)-bc)x+b^(2)-ac=0 are real and equal prove that either a=0 (or) a^(3)+b^(3)+c^(3)=3"abc" .

If the roots of (a-b)x^(2)+(b-c)x+(c-a)=0 are real and equal, then prove that b, a, c are in arithmetic progression.

If a ,b ,c in R such that a+b+c=0a n da!=c , then prove that the roots of (b+c-a)x^2+(c+a-b)x+(a+b-c)=0 are real and distinct.

If the roots of the equation a(b-c)x^2+b(c-a)x+c(a-b)=0 are equal, show that 2//b=1//a+1//c dot

The equation a x^2+b x+c=0 has real and positive roots. Prove that the roots of the equation a^2x^2+a(3b-2c)x+(2b-c)(b-c)+a c=0 re real and positive.

If a, b are real then show that the roots of the equation (a-b)x^(2)-6(a+b)x-9(a-b)=0 are real and unequal.

Let a , ba n dc be the three sides of a triangle, then prove that the equation b^2x^2+(b^2=c^2-alpha^2)x+c^2=0 has imaginary roots.