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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 ?

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    http://www.math-only-math.com/condition-for-common-root-or-roots-of-quadratic-equations.html – 2017-01-31
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    Do you know the resultant of two polynomials? – 2017-01-31

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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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    How you have eliminated x – 2017-01-31
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    @koolman $x^2 =(x)^2$ – 2017-01-31