Question

Form the equation whose roots are $1\pm \sqrt{-2},2\pm \sqrt{-3}$.

Answer 1

$1\pm \sqrt{-2}=1\pm \sqrt{2}i;2\pm \sqrt{-3}=2\pm \sqrt{3}i$

The equation with given roots is

$\{x-(1+\sqrt{2}i\left)\right\}\{x-(1-\sqrt{2}i\left)\right\}\{x-(2+\sqrt{3}i\left)\right\}\{x-(2-\sqrt{3}i\left)\right\}=0\left\{\right(x-1)-\sqrt{2}i\}\left\{\right(x-1)+\sqrt{2}i\}\left\{\right(x-2)-\sqrt{3}i\}\left\{\right(x-2)+\sqrt{3}i\}=0\{{(x-1)}^2-{\left(\sqrt{2}i\right)}^2\}\{{(x-2)}^2-{\left(\sqrt{3}i\right)}^2\}(x^2-2x+3)(x^2-4x+7)=0x^4-4x^3+7x^2-2x^3+8x^2-14x+3x^2-12x+21=0x^2-6x^3+18x^2-26x+21=0$