## Task B-4.3.

The vertices of the quadrilateral are points in the complex plane associated with the solutions of the equation

$$
\left(z^{2}+1\right)\left(z^{2}+z+1\right)=0
$$

in the set $\mathbb{C}$. Determine the coordinates of the vertices and the area of the given quadrilateral.
## Solution.

The solutions we seek are obtained by solving the equations $z^{2}+1=0$ and $z^{2}+z+1=0$.

The solutions to the first equation are $z_{1}=i, z_{2}=-i$.

The solutions to the second equation are $z_{3}=-\frac{1}{2}+\frac{\sqrt{3}}{2}i$ and $z_{4}=-\frac{1}{2}-\frac{\sqrt{3}}{2}i$.

The vertices of the quadrilateral are points $A(0,-1), B(0,1), C\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right), D\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$.

![](https://cdn.mathpix.com/cropped/2024_05_30_d455efeee432fadf0574g-19.jpg?height=631&width=605&top_left_y=1506&top_left_x=637)

The area of the obtained trapezoid is $P=\frac{2+\sqrt{3}}{2} \cdot \frac{1}{2}=\frac{2+\sqrt{3}}{4}$ square units.

1 point