Explanation

(d)
Let $D_{1}\ and\ D_{2}$ be discriminants of $x^{2} + b_{1}x + c_{1} = 0$ and $x^{2} + b_{2}x + c_{2} = 0$ respectively. Then
$D_{1} + D_{2} = b_{1}^{2} - 4c_{1} + b_{2}^{2} - 4c_{2}$
$= \left( b_{1}^{2} + b_{2}^{2} \right) - 4(c_{1} + c_{2})$
$= b_{1}^{2} + b_{2}^{2} - 2b_{1}b_{2}\ \ \ \$ [$\because b_{1}b_{2} = 2\left( c_{1} + c_{2} \right),\$given]
$= \left( b_{1}{- b}_{2} \right)^{2} \geq 0$
$\Rightarrow D_{1} \geq 0\ or\ {\ D}_{2} \geq 0$
$\Rightarrow D_{1}\ and\ D_{2}$ both are positive.