What are the roots of $x^4+4=0$?

Given: Equation: $x^4+4=0$.

To do: To find the roots of the equation:  $x^4+4=0$.

$x^4+4=0$

$\Rightarrow x^4+4+4x^2−4x^2=0$

$\Rightarrow x^4+4x^2+4−4x^2=0$

$\Rightarrow ( x^2)^2+2.2.x^2+2^2−(2x)^2=0$

$\Rightarrow ( x^2+2)^2−(2x)^2=0$

$\Rightarrow ( x^2+2−2x)(x^2+2+2x)=0$

$\Rightarrow x^2−2x+2=0,\ x^2+2x+2=0$

$\Rightarrow x=\frac{2\pm\sqrt{4-8}}{2},\ x=\frac{-2\pm\sqrt{4-8}}{2}$

$\Rightarrow x=\frac{2\pm\sqrt{-4}}{2},\ x=\frac{-2\pm\sqrt{-4}}{2}$

$\Rightarrow x=\frac{2\pm\sqrt{4( -1)}}{2},\ x=\frac{-2\pm\sqrt{4( -1)}}{2}$

$\Rightarrow x=\frac{2\pm2\sqrt{( -1)}}{2},\ x=\frac{-2\pm2\sqrt{( -1)}}{2}$

$\Rightarrow x=\frac{2\pm2i}{2},\ x=\frac{-2\pm2i}{2}$

$\Rightarrow x=2( \frac{1\pm i}{2}),\ x=2( \frac{-1\pm i}{2})$

$\Rightarrow x=1\pm i,\ x=-1\pm i$

$\therefore x=1+i,\ 1−i,\ −1+i,\ −1−i$ are the roots of $x^4+4=0$.