If $x^2-cx+d=0,x^2-ax+b=0$  have one common root and second has equal roots then $2(b+d)$  is 

$x^2-cx+d=0$

$\alpha ,\beta$

$\alpha +\beta =c,\alpha \beta =d$

$x^2-cx+d=0$

$x^2-ax+b=0$

$\alpha ,\alpha$

$\alpha +\alpha =a,{\alpha }^2=b$

$2\alpha =a\Rightarrow \alpha =\frac{a}{2}$

$\alpha$is the root of the above equation

${\alpha }^2-c\alpha +d=0$

$b-c\left(\frac{a}{2}\right)+d=0$

$b-\frac{ac}{2}+d=0$

$2(b+d)=ac$