Let a,b,c be rational, consider the equation $\mathrm{a}(\mathrm{b}-\mathrm{c}){\mathrm{x}}^2+\mathrm{b}(\mathrm{c}-\mathrm{a})\mathrm{x}+\mathrm{c}(\mathrm{a}-\mathrm{b})=0$. Then

The given equation is$\mathrm{a}(\mathrm{b}-\mathrm{c}){\mathrm{x}}^2+\mathrm{b}(\mathrm{c}-\mathrm{a})\mathrm{x}+\mathrm{c}(\mathrm{a}-\mathrm{b})=0$

Since the $\text{sum of coefficients }=0$

$\Rightarrow \mathrm{x}=1\text{ is a root }$

The other root is $\frac{c\left(a-b\right)}{a\left(b-c\right)}$

If the roots are equal $\frac{c\left(a-b\right)}{a\left(b-c\right)}=1$

It implies that 

        $\begin{array}{rcl}ca-bc& =& ab-ac\\ 2ac& =& ab+bc\\ \frac{2}{b}& =& \frac{1}{a}+\frac{1}{c}\end{array}$

If $a=b=c$ then the equation is an identity. so it has infinitely many solution