1971 Canadian MO Problems/Problem 4

Problem

Determine all real numbers $\displaystyle a$ such that the two polynomials $\displaystyle x^2+ax+1$ and $\displaystyle x^2+x+a$ have at least one root in common.

Solution

Let this root be $\displaystyle r$ . Then we have

$\displaystyle \begin{matrix} r^2 + ar + 1 &=& r^2 + r + a\\ ar + 1 &=& r + a\\ (a-1)r &=& (a-1)\end{matrix}$

Now, if $\displaystyle a = 1$ , then we're done, since this satisfies the problem's conditions. If $\displaystyle a \neq 1$ , then we can divide both sides by $\displaystyle (a - 1)$ to obtain $\displaystyle r = 1$ . Substituting this value into the first polynomial gives

$\begin{matrix} 1 + a + 1 &=& 0\\ a &=& -2 \end{matrix}$

It is easy to see that this value works for the second polynomial as well.

Therefore the only possible values of $\displaystyle a$ are $\displaystyle 1$ and $\displaystyle -2$ . Q.E.D.

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1971 Canadian MO Problems/Problem 4

Problem

Solution