<s>[INST] If each pair of equations $x^2=b_1x+c_1=0,x^2=b_2x+c_2 \text{ and } x^2+b_3x=c_3$ have a common root, prove following If each pair of equations $x^2=b_1x+c_1=0,x^2=b_2x+c_2 \text{ and } x^2+b_3x=c_3$ have a common root, prove that
$(b_1+b_2+b_3)^2=4(c_1+c_2+c_3+b_1b_2)$
My attempt is as follows:
For equations $x^2=b_1x+c_1,x^2=b_2x+c_2$ to have a common root:
$(c_2-c_1)^2=(b_1c_2-b_2c_1)(b_1-b_2)$
For equations $x^2=b_2x+c_2,x^2+b_3x=c_3$ to have a common root:
$(c_3-c_2)^2=(b_2c_3+b_3c_2)(b_3+b_2)$
For equations $x^2=b_1x+c_1,x^2+b_3x=c_3$ to have a common root:
$(c_3-c_1)^2=(b_1c_3+b_3c_1)(b_1+b_3)$
Adding all three equations:
$2({c_1}^2+{c_2}^2+{c_3}^2-c_1c_2-c_2c_3-c_3c_1)=({b_1}^2+b_2b_3)(c_2+c_3)+({b_2}^2+b_1b_3)(c_1+c_3)+({b_3}^2-b_1b_2)(c_1+c_2)$
But from here I was not able to proceed towards the proof. Please help me in this.
 [/INST] I finally got this, thanks to @lab bhattacharjee
There are three quadratic equations 
\begin{equation}
x^2-b_1x-c_1=0\tag{1}
\end{equation}
\begin{equation}
x^2-b_2x-c_2=0\tag{2}
\end{equation}
\begin{equation}
x^2+b_3x-c_3=0\tag{3}
\end{equation}
Suppose $(1)$ and $(2)$ have a common root as $p$, $(2)$ and $(3)$ have a common root as $q$, (1) and (3) have a common root as $r$.
\begin{equation}
p+r=b_1\tag{4}
\end{equation}
\begin{equation}
p+q=b_2\tag{5}
\end{equation}
\begin{equation}
-(q+r)=b_3\tag{6}
\end{equation}
$$b_1+b_2+b_3=2p$$
\begin{equation}
(b_1+b_2+b_3)^2=4p^2\tag{7}
\end{equation}
\begin{equation}
-pr=c_1\tag{8}
\end{equation}
\begin{equation}
-pq=c_2\tag{9}
\end{equation}
\begin{equation}
-qr=c_3\tag{10}
\end{equation}
Adding $(8),(9),(10)$
\begin{equation}
4(c1+c2+c3)=-4(pq+qr+pr)\tag{11}
\end{equation}
\begin{equation}
4b_1b_2=4(p+r)(p+q)\tag{12}
\end{equation}
Adding $(11),(12)$ 
$$4(c_1+c_2+c_3+b_1b_2)=-4(pq+qr+pr)+4(p^2+pq+pr+qr)$$
$$4(c_1+c_2+c_3+b_1b_2)=4p^2$$
Hence $4(c_1+c_2+c_3+b_1b_2)=(b_1+b_2+b_3)^2$
</s>