predicting the nature of roots of the equation

Question

Let $$bcx^2+2acx+ab=0$$ $$acx^2+2abx+bc=0$$ $$abx^2+2bcx+ac=0$$ be three quadratic equations.where $a,b,c$ are real numbers such that a,b,c are not all equal.If $d$ is a common root of the three equations then the question is to predict the nature of roots of third equation and prove that $d<0$ .

It is clear that the common root is real.For if it were imaginary then its conjugate would also be a common root implying $a=b=c$.I couldn't proceed after this.Please help me in this regard.Thanks.

Answer 1

Add all the three equations: $$ (ab+ac+bc)(x^2 + 2x+1) = 0 $$

Suppose that $d$ is a common root of each of the three equations, then it is a root of this equation also, but then $(d+1)^2 = 0 \implies d=-1$ follows ,unless $ab+ac+bc = 0$, this happens whenever $c = \dfrac{-ab}{a+b}$, then you substitute for $c$ and solve the remaining equations (I leave you to do that, it is just quadratic formula).