Task B-3.6. Determine all real numbers $x$ and $y$ for which

$$
5 \cdot \cos ^{2} y+x^{2}-2 x \cdot \cos y-8 x+20=0
$$
First solution. Let's solve the given equation as a quadratic equation in $x$.

$$
\begin{gathered}
x^{2}-2 x \cdot(\cos y+4)+20+5 \cdot \cos ^{2} y=0 \\
x_{1,2}=\frac{2 \cdot(\cos y+4) \pm \sqrt{4 \cdot\left(\cos ^{2} y+8 \cos y+16\right)-4 \cdot\left(20+5 \cos ^{2} y\right)}}{2} \\
x_{1,2}=\cos y+4 \pm \sqrt{-4 \cdot \cos ^{2} y+8 \cos y-4} \\
x_{1,2}=\cos y+4 \pm \sqrt{-(2 \cos y-2)^{2}}
\end{gathered}
$$

The solution will be real only if the discriminant is zero, i.e., $\cos y=1$.

Then $x=5$.

The solution is $x=5, y=2 k \pi, k \in \mathbb{Z}$.