Let w be a complex number such that |w| = 1, and the equation has a pure imaginary root z^2 + z = w. Find |z|
+397 \(z^{2}+z= w,\\ |z^{2}+z|=|w| =1.\)
z is to be purely imaginary, so let z = iy, where y is a real number.
Then,
\(|-y^{2}+iy|=1,\\ \sqrt{y^{4}+y^{2}}=1,\\ y^{4}+y^{2}-1=0,\\ \displaystyle y^{2}=\frac{-1 \pm\sqrt{5}}{2}\)
y is to be a real number, so the negative sign is rejected, and then
\(\displaystyle y = \pm\sqrt{\frac{\sqrt{5}-1}{2}}\)
so,
\(\displaystyle |z|=\sqrt{\frac{\sqrt{5}-1}{2}}.\)
