# Problem 6. Maximum 15 points

Find the values of the variable $x$ such that the four expressions: $2 x-6, x^{2}-4 x+5, 4 x-8, 3 x^{2}-12 x+11$ differ from each other by the same number. Find all possible integer values of the variable $x$ for any order of the expressions.
# Solution:

From the properties of numbers that differ from the following in a numerical sequence by the same number, we form two equations:

$$
\begin{gathered}
x^{2}-4 x+5-(2 x-6)=4 x-8-\left(x^{2}-4 x+5\right) \\
4 x-8-\left(x^{2}-4 x+5\right)=3 x^{2}-12 x+11-(4 x-8)
\end{gathered}
$$

Only \( x=4 \) satisfies both equations.

## Criteria.

- A correct solution to the problem and the correct answer are provided - 15 points (maximum),
- A correct solution to the problem is provided, but the wrong answer or no answer is given - 8 points (rounded to the nearest whole number, half the maximum),
- An error in the solution, but the correct answer is obtained - 7 points (rounded to the nearest whole number, half the maximum minus one),
- The solution is incorrect or missing, but the correct answer is obtained - 1 point (under any circumstances),
- The solution is incorrect or missing, and the wrong answer or no answer is given - 0 points.