<s>[INST] Prove relations between the roots of 3 quadratic equations 
Let $x_1, x_2$ be the roots of the equation $x^2 + ax + bc = 0$, and $x_2, x_3$ the roots of
  the equation $x^2 + bx + ac = 0$ with $ac \neq bc$. Show that $x_1, x_3$ are the roots of the
  equation $x^2 + cx + ab=0$.

From Vieta's I have:
$\begin{cases} x_1+x_2=-a\\x_1x_2=bc\end{cases}$ $\begin{cases} x_2+x_3=-b\\x_2x_3=ac\end{cases}$
and I have to prove:
 $\begin{cases} x_1+x_3=-a\\x_1x_3=ab\end{cases}$
 [/INST] Subtract the first two equations to get $x_2$

 $$\begin{align}x_2^2+ax_2+bc&=0\\x_2^2+bx_2+ac&=0\end{align}$$ to get $$(a-b)x_2-(a-b)c=0$$ from where $x_2=c$.
 So, $$\begin{align}x_1&=b=-a-c\\x_3&=a=-b-c\end{align}$$ from where $$\begin{align}x_1+x_3&=a+b=c\\x_1x_3&=ab\end{align}$$

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