If $c$ and $d$ are the roots of $x^2+a x+b=0$, then a root of $x^2+(4 c+a) x+\left(b+2 a c+4 c^2\right)=0$ is

Question

Aicharya · Accepted Answer

Given that $c$ and $d$ are the roots of $x^2+a x+b=0$ $\begin{aligned} &\Rightarrow c+d=-a \& c d=b ... (i)\ &\begin{aligned} & 	ext { Now, } x^2+(4 c+a) x+\left(b+2 a c+4 c^2ight)=0 \ & \Rightarrow x^2(4 c-c-d) x+\left(b-2 c(c+d)+4 c^2ight)=0 \ & \Rightarrow x^2+(3 c-d) x+2 c^2-c d=0 \ & \Rightarrow x^2+c x+(2 c-d) x+c(2 c-d)=0 \ & \Rightarrow(x+c)(x+2 c-d)=0 \ & \Rightarrow x=d-2 c . \end{aligned} \end{aligned}$

If $c$ and $d$ are the roots of $x^2+a x+b=0$, then a root of $x^2+(4 c+a) x+\left(b+2 a c+4 c^2\right)=0$ is

Solution