The equations $x^2+2x+k=0$ and $x^2+kx+2=0$, with $k$ ≠ 2, share a common root. That root is:

Subtracting removes $x^2:(2-k)x+(k-2)=0$, i.e. $(2-k)x=(2-k)$. Since $k$ ≠ 2 we may divide by $(2-k)$ to get $x=1$. (−3 is the resulting value of $k$, found from $1+2+k=0$, not the root; and $k=2$ is excluded precisely because the subtraction then collapses to 0 = 0.) Check with $k=-3:x^2+2x-3=0$ and $x^2-3x+2=0$ both vanish at $x=1$.