If a and b are positive numbers, prove that the equation `1/(x-a)+1/(x-b)+1/x=0` has two real roots one between `a/3` and `2a/3` and the other between `-2b/3` and `-b/3`

If a and b are positive numbers,prove that the equation (1)/(x-a)+(1)/(x-b)+(1)/(x)=0 has two real roots one between (a)/(3) and 2(a)/(3) and the other between -2(b)/(3) and -(b)/(3)

If a , b , c , are real, then prove that roots of the equation 1/(x-a) + 1/(x-b) + 1/(x-c) = 0 are real.

If bgta , then prove that the equation (x-a)(x-b)-1=0 has one root in (-infty,a) and the other in (b,+infty) .

If a and b are positive real numbers and each of the equations x^(2)+3ax+b=0 and x^(2)+bx+3a=0 has real roots,then the smallest value of (a+b) is

If the equation x^2-b x+1=0 does not possess real roots, then -3 2 (d) b<-2

if one of the root of the equation a(b-c)x^(2)+b(c-a)x+c(a-b)=0 is 1,the other root is

If the equation a x^2+b x+c=0 has two positive and real roots, then prove that the equation a x^2+(b+6a)x+(c+3b)=0 has at least one positive real root.

If the equation a x^2+b x+c=0 has two positive and real roots, then prove that the equation a x^2+(b+6a)x+(c+3b)=0 has at least one positive real root.

If b^2<2a c , then prove that a x^3+b x^2+c x+d=0 has exactly one real root.

If b^2<2a c , then prove that a x^3+b x^2+c x+d=0 has exactly one real root.