Find the real number $k$ for which $1+ki$ (where $i=\sqrt{-1}$) is a zero of the polynomial $x^2+kx+5$

Question

I expanded the factors $(x-(1+ki))$ and $(x-(1-ki))$ to get $x^2-2x-k^2+1$, so $k=-2$

Is this answer correct? Because in my book it says $k=6$.

Answer 1

By Vieta formula $x_1+x_2= -{b\over a}$ where $x_1,x_2$ are the solution of quadratic equation $ax^2+bx+c=0$ we get $$1+ki+1-ki = -k$$ so $k=-2$.

Answer 2

or you plug in $$1+ki$$ in your equation $$(1+ki)^2+k(1+ki)+5=0$$