If the equations $ax^2+2bx+c=0$ and $a_1x^2+2b_1x+c_1=0$ have one and only one root common, then prove that $b^2-ac$ and $b_1^2-a_1c_1$ are perfect squares.
Attempt at a solution:-
Let the common root be $\alpha$. Then $x=\alpha$ must satisfy both the quadratic equations. Hence, we have
On solving $(1)$ and $(2)$, we get
From this, we get $$(a_1c-ac_1)^2=(2ab_1-2a_1b)(2bc_1-2b_1c)$$
On expanding and manipulating this I couldn't get anywhere close to what the question needs me to prove.
If you can think of an intuitive solution using graphs of the quadratic equations then it would be helpful too.