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I have posted a link there to this topic so the OP can see my work.Write a quadratic equation with the solution set -1 + i√2, -1 - i√2?

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I have posted a link there to this topic so the OP can see my work.Write a quadratic equation with the solution set -1 + i√2, -1 - i√2?

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I would begin by writing the roots as:

\(\displaystyle x=-1\pm i\sqrt{2}=\frac{-2\pm\sqrt{-8}}{2}\)

And so from the quadratic formula:

\(\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)

we see we have:

\(\displaystyle a=1,\,b=2,\,b^2-4ac=4-4c=-8\implies c=3\)

Hence, the quadratic:

\(\displaystyle f(x)=x^2+2x+3\)

has the given roots.