Question

If z be a complex number satisfying $z^2$ + z + 1 = 0. Find the value of $\left|z\right|$.

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Answer 1

Let z = x + iy

Substitute z in given equation

$x^2$ - $y^2$ + 2xyi + x + iy +1 = 0

($x^2$ - $y^2$ + x +1) + y(2x + 1)1 = 0

Comparing real and imaginary part of the above given equation

$x^2$ - $y^2$ + x +1 = 0----------------(1) or

y(2x + 1) = 0 ---------(2)

y = 0,2x + 1=0

$$\begin{array}{cc}Substitutevalueofyinequation1& When2x+1=0,x=\frac{-1}{2}\\ wheny=0& \frac{1}{4}-y^2+\frac{1}{2}=0\\ x^2+x+1=0& y^2=\frac{3}{4}\\ x=\frac{-1\underset{–}{+}\sqrt{-3}}{2}& y=\frac{\underset{–}{+}\sqrt{3}}{2},x=\frac{-1}{2}\\ x=\frac{-1\underset{–}{+}\sqrt{3i}}{2}& \end{array}$$

Substitute the value of x and y to get the complex number z.

Case 1: when y = 0

Complex number

z = x + iy

= $\frac{-1\underset{–}{+}\sqrt{3i}}{2}$

Case 2: when x = $\frac{-1}{2}$, y = $\frac{\underset{–}{+}\sqrt{3}}{2}$

z = x + iy

= $\frac{-1\underset{–}{+}\sqrt{3i}}{2}$

$\left|z\right|$ = $\sqrt{\frac{1}{4}+\frac{3}{4}}$ = $\sqrt{1}$ = 1

So, modulus of z = 1.