<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 then, consider $y^2+by+c=0$ and $by^2+cy+1=0.$ Clearly $y\not=0.$ Therefore $$y^3+by^2+cy=0$$
$$y^3-1=0$$
$$(y-1)(y^2+y+1)=0.$$
If $y=1,$ then clearly $b+c+1=0.$
If $y^2+y+1=0,$ then note that $by+c=y+1$ and $cy+1=b(y+1).$ Solve this two simple linear equations and equate the value of $y.$
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