<s>[INST] Prove the given condition from given two quadratic equation 
Question: If the quadratic equations $x^2+bx+c=0$ and $bx^2+cx+1=0$ have a
  common root then prove that either $b + c + 1 = 0$ or $b^2 + c^2 + 1 =bc + b + c$

Till yet,
I had figured the common root of the given two quadratic equation. i.e.
Multiplying first equation by $b$ and eliminating the term $bx^2$ from the equation I get the common root ($\alpha$ say),
$$\alpha=\frac{1 - cb}{b^2 - c}$$
Further putting this value in either of the equation didn't benefited me much.What it gave me was an odd, unfriendly equation. Can anyone help me in this?
Thanks in advance.
 [/INST] Let $y$ be the common root
So, we have $$y^2+by+c=0, by^2+cy+1=0$$
Solving for $y,y^2$ we get $y^2=\dfrac{b-c^2}{c-b^2},y=\dfrac{bc-1}{c-b^2}$
and using the identity $y^2=(y)^2$ we get $$b^3+c^3+1^3-3\cdot b\cdot c\cdot1=0$$
Now use Factorize the polynomial $x^3+y^3+z^3-3xyz$
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