## Task A-2.7. (10 points)

Determine and sketch in the complex plane the set of all numbers $z$ that satisfy the condition

$$
\operatorname{Re}\left[(4+3 i) z^{2}\right] \geqslant 0
$$
## First Solution.

By substituting $z=x+i y$, we get

$$
\begin{aligned}
(4+3 i) z^{2} & =(4+3 i)(x+i y)^{2} \\
& =4 x^{2}-4 y^{2}-6 x y+i\left(3 x^{2}-3 y^{2}+8 x y\right) \\
\operatorname{Re} & \left.(4+3 i) z^{2}\right]=4 x^{2}-4 y^{2}-6 x y
\end{aligned}
$$

We need to determine the set of all pairs $(x, y)$ of real numbers that satisfy the condition

$$
4 x^{2}-4 y^{2}-6 x y \geqslant 0
$$

By rearranging, we get:

$$
\begin{gathered}
4 x^{2}-6 x y+\frac{9}{4} y^{2} \geqslant 4 y^{2}+\frac{9}{4} y^{2} \\
\left(2 x-\frac{3}{2} y\right)^{2} \geqslant\left(\frac{5}{2} y\right)^{2}
\end{gathered}
$$

from which it follows:

$$
(x-2 y)(2 x+y) \geqslant 0
$$

The desired set is a region in the plane bounded by the lines $x-2 y=0$ and $2 x+y=0$ 1 point as shown in the figure:

![](https://cdn.mathpix.com/cropped/2024_05_30_13589bc3b5dd31895527g-10.jpg?height=554&width=986&top_left_y=1448&top_left_x=455)