if a, b,c are real numbers and not all equal, prove that the equation

Question

(c-a)x^2 -2(a-b)x + (b+c) = 0 has unequal real roots

Answer 1

if the discriminant is positive, the roots are real and unequal. The discriminant is

4(a-b)^2 - 4(c-a)(b+c)
= 4(a^2-2ab+b^2 + ab+ac-bc-c^2)

Hmmm. What if a=b? we have a^2-c^2.
So, if a^2 < c^2 the roots are complex.

well, maybe you mean that a,b,c are all nonzero. In that case, we still have a problem, since we can let a be almost equal to b, and we have, for example,

a=2 b=2.01 c=3 the discriminant is -20.04 and the roots are complex.