Solving Quadratic Equations with Complex Numbers

Solve x^{2}-2x+5=0 over the complex numbers.

x^{2}-2x+5=0

quadratic\ formula: x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}

a=1

b=-2

c=5

x=\frac{-(-2)\pm\sqrt{(-2)^{2}-4(1)(5)}}{2(1)}

x=\frac{2\pm\sqrt{4-20}}{2(1)}

x=\frac{2\pm\sqrt{-16}}{2}\rightarrow \sqrt{-16}=\sqrt{-1}\sqrt{16}=\sqrt{i^2}\sqrt{16}=i\sqrt{16}

x=\frac{2\pm\sqrt{16}i}{2}

x=\frac{2\pm 4i}{2}

x=1\pm 2i

x=1+2i,\ 1-2i