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# Complex numbers

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Let w be a complex number such that |w| = 1, and the equation has a pure imaginary root z^2 + z = w. Find |z|

Nov 16, 2022

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$$z^{2}+z= w,\\ |z^{2}+z|=|w| =1.$$

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,

$$\displaystyle |z|=\sqrt{\frac{\sqrt{5}-1}{2}}.$$

Nov 17, 2022