If $x^2 + mx + 1 = 0$ and $x^2 + x + m = 0$ share a same root ($m \neq 1$), find $m$

Question

I'm certain there must be a certain formula/theorem involved but I haven't learnt about it. Rather than simply being given the answer, could I be given short and concise useful tips for questions like these in the future? Much thanks in advance.

Answer 1

Let the common root be $x$ then

$ x^2 + m x + 1 = 0 $ and $ x^2 + x + m = 0 $

Hence, by subtracting

$ m x + 1 - (x + m) = 0 $

$ (m - 1) x + (1 - m) = 0 $

Since $ m \ne 1 $ then $ x = 1 $ is the common root.

Substituting $x = 1$ in either equation gives $ m = -2 $

Answer 2

There are a set of formulas named Viête formulas: You can use those.

In this case, if $x_1$ and $x_2$ are the roots of the first equation and $x_1$ and $x_3$ the roots of the second one (note that $x_1 is the common solution of both equations).

For the first equation the Viête formulas are $x_1+x_2=-m$ and $x_1x_2=1$. For the second equations are $x_1+x_3=-1$ and $x_1x_3=m$.

In other words, you can solve this system of equations for $x_1$, $x_2$ and $x_3$.

Does this help? :D