Question

Prove that the roots of the equation `x^4-2x^2+4=0` forms a rectangle.

Answer 1

`x^(4)-2x^(2)+4=0`
`:." "x^(2)=(2+-sqrt(4-16))/(2)=1+-sqrt(3)i=2(1+-sqrt(3)i)/(2)`
`=2(cos(pi)/(3)+-isin(pi)/(3))`
`:." "x=+-sqrt(2)(cos(pi)/(3)+-isin(pi)/(3))^(1//2)`
`=+-sqrt(2)(cos(pi)/(6)+-isin(pi)/(6))`
`:." "x=sqrt(2)e^(ipi//6),-sqrt(2)e^(ipi//6),-sqrt(2)e^(-ipi//6)`
Clearly these points forms rectangle inscribed in circle of radius `sqrt(2)`.