Maths Olympiad Prep

Library / /125 of 520

Algebra Difficulty 2.2 Junior Find the answer

For how many values of the coefficient a do the equations x2+ax+1=0x2−x−a=0\begin{align*}x^2+ax+1=0 \\ x^2-x-a=0\end{align*} have a common real solution?

Pick one

Solution

Subtracting the equations, we get ax+x+1+a=0ax+x+1+a=0, or (x+1)(a+1)=0(x+1)(a+1)=0, so x=−1x=-1 or a=−1a=-1. If x=−1x=-1, then a=2a=2, which satisfies the condition. If a=−1a=-1, then xx is nonreal. This means that a=−1a=-1 is the only number that works, so our answer is (B)(B).
~alexanderruan

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.