IF a,b,c are real, prove that the roots of the equation `1/(x-a)+1/(x-b)+1/(x-c)=0` are always real and cannot be equal unless a=b=c.

If a,b, are real, then show that the roots of the equation (x-a)(x-b) = b^2 are always real.

Prove that both the roots of the equation (x-a)(x-b)+ (x-b)(x-c)+(x-a)(x-c)=0 are real.

IF a,b,c are real, show that the roots of each of the following equations are real: 1/(x-a)+1/(x-b)+1/(x-c)=0

If a, b, c are real, then both the roots of the equation (x -b )(x -c)+(x -c)(x - a)+(x - a)(x - b)=0 are always

If a ,b ,c are distinct positive numbers, then the nature of roots of the equation 1/(x-a) + 1/(x-b) + 1/(x-c) = 1/x is a) all real and is distinct b) all real and at least two are distinct c) at least two real d) all non-real

IF a,b, are real, show that the roots of each of the following equations are real: 1/(x-a)+1/(x-b)=1/a^2

IF a,b,c are real, show that the roots of each of the following equations are real: (x-a)(x-b)=b^2

Prove that, if the roots of the equation (a^2+b^2)x^2+2(bc+ad)x+(c^2+d^2)=0 be real, then they are equal.

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

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