## Task A-2.1.

Determine all real numbers $c$ for which one solution of the quadratic equation

$$
27 x^{2}-12 x+c=0
$$

is the square of the other solution.
## Solution.

Let $b$ and $b^{2}$ be the solutions of the given equation.

According to Viète's formula for the sum of the solutions, we have $b+b^{2}=\frac{12}{27}$.

The solutions of this equation are $b_{1}=\frac{1}{3}$ and $b_{2}=\frac{-4}{3}$.

According to Viète's formula for the product of the solutions, we have $b \cdot b^{2}=\frac{c}{27}$, i.e., $c=27 b^{3}$.

Therefore, we have $c_{1}=1$ and $c_{2}=-64$.