Animated Solution for Mathematics - Complex Numbers: Let $S=\{z\in \mathbb{C}:4z^2+\bar{z}=0\}$. Then ${\sum }_{z\in S}|z{|}^2$ is equal to:

To solve the equation $4z^2+\bar{z}=0$, we begin by expressing the complex number $z$ in its Cartesian form. Let $z=x+iy$, where $x,y\in \mathbb{R}$.

Consequently, the conjugate is $\bar{z}=x-iy$. Substituting these into the original equation, we obtain:

Expanding the squared term, we have $4(x^2-y^2+2ixy)+x-iy=0$. Distributing the constant, we get:

Now, we separate the expression into its real and imaginary components:

For the complex number to be zero, both the real and imaginary parts must vanish independently. This gives us a system of two equations:

1. Real part: $4x^2-4y^2+x=0$ 2. Imaginary part: $y(8x-1)=0$

The imaginary part equation, $y(8x-1)=0$, implies two distinct cases: $y=0$ or $x=\frac{1}{8}$.

Substituting $y=0$ into the real part equation:

This yields two solutions: $x=0$ and $x=-\frac{1}{4}$. The corresponding complex roots are $z_1=0$ and $z_2=-\frac{1}{4}$.

Substituting $x=\frac{1}{8}$ into the real part equation $4x^2-4y^2+x=0$:

This yields two roots where $y=\pm \frac{\sqrt{3}}{8}$. Thus, $z_3=\frac{1}{8}+i\frac{\sqrt{3}}{8}$ and $z_4=\frac{1}{8}-i\frac{\sqrt{3}}{8}$.

We calculate the squared magnitude $|z{|}^2=x^2+y^2$ for each root:

For $z_1,z_2$: $|z_1{|}^2=0$ and $|z_2{|}^2=(-\frac{1}{4}{)}^2=\frac{1}{16}$.

For $z_3,z_4$: $|z{|}^2=(\frac{1}{8}{)}^2+(\frac{\sqrt{3}}{8}{)}^2=\frac{1}{64}+\frac{3}{64}=\frac{4}{64}=\frac{1}{16}$.

Summing the squared magnitudes of all roots: