Question

What is the S to the 10-th power, if S is the sum of the solutions of the equation 2x2 - 2ix +10=0, where i=√(-1).

Answer 1

`2x^2-2ix+10=0\implies x=(-(-2i)\pm√((-2i)^2-4*2*10))/(2*2)=\frac{2i\pm√(-84)}4`

When you add the roots, the square root terms will cancel, leaving you with


`S=\frac{2i}4+\frac{2i}4=\frac{4i}4=i`

and so
`S^(10)=i^(10)=(i^4)^2i^2=-1`.