8-4. Find all common points of the graphs

$$
y=8 \cos \pi x \cdot \cos ^{2} 2 \pi x \cdot \cos 4 \pi x \quad \text { and } \quad y=\cos 5 \pi x
$$

with abscissas belonging to the interval $[-1 ; 0]$. In your answer, specify the sum of the abscissas of the found points.
Solution. This is equivalent to $\sin 10 \pi x=\sin (-4 \pi x), x \neq n, n \in \mathbb{Z}$. From this, $x=k / 7$ or $x=1 / 6+l / 3, k, l \in \mathbb{Z}$.

The roots of the equation in the interval $[-1 ; 0]$ are: $-1 / 7, -2 / 7, -3 / 7, -4 / 7, -5 / 7, -6 / 7$ and $-1 / 6, -1 / 2$, $-5 / 6$.

Answer: $-4.5$.