<s>[INST] Common roots of a quadratic equation If the quadratic equations, $ x^2
 + bx + c = 0$ and $ bx^2
 + cx + 1 = 0 $ have a common root then we have to prove that either $b + c + 1 = 0$ or $ b^2
 + c^2 + 1 = bc + b + c$.
I tried it a lot , but not able to proceed further . 
Can anybody provide me a hint ? 
 [/INST] We have that both these equations have a common root. Solving these equations for $x^2$ and $x $, we get, $$x^2=\frac {b-c^2}{c-b^2} \text {   and  } x =\frac {bc-1}{c-b^2} $$ Now eliminating $x $, we get, $$b^3+c^3+1-3bc=0$$ $$\Rightarrow (b+c)^3-3bc (b+c) +1 -3bc =0$$ $$\Rightarrow (b+c+1)(b^2-(c+1)b +(c^2-c+1))=0$$ and the result follows. Hope it helps. 
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