### Instruction: Determine the sum of all real numbers $x$ satisfying
\[(x^2-4x+2)^{x^2-5x+2} = 1.\].### Response: This problem involves an equation where the base and the exponent are both variable expressions.
One way to approach this problem is to think about the cases where an exponential expression can equal 1.
I know that any number raised to the zero power is 1, so if the exponent $x^2-5x+2$ is zero, then the equation will be true.
I also know that 1 raised to any power is 1, so if the base $x^2-4x+2$ is 1, then the equation will also be true.
I will try to find the values of $x$ that satisfy either of these cases.
To find the values of $x$ that make the exponent zero, I need to solve the quadratic equation $x^2-5x+2=0$.
I can use the quadratic formula to find the solutions of this equation.
The quadratic formula says that the solutions of $ax^2+bx+c=0$ are given by $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
In this case, $a=1$, $b=-5$, and $c=2$, so I can plug these values into the formula and simplify.
I get $x=\frac{5\pm\sqrt{25-8}}{2}=\frac{5\pm\sqrt{17}}{2}$.
These are two real numbers that make the exponent zero and therefore make the equation true.
To find the values of $x$ that make the base 1, I need to solve the quadratic equation $x^2-4x+2=1$.
I can subtract 1 from both sides to get $x^2-4x+1=0$.
I can also use the quadratic formula to find the solutions of this equation.
The solutions are given by $x=\frac{4\pm\sqrt{16-4}}{2}=\frac{4\pm\sqrt{12}}{2}$.
I can simplify the square root of 12 by factoring out a 4.
I get $x=\frac{4\pm 2\sqrt{3}}{2}=2\pm\sqrt{3}$.
These are two more real numbers that make the base 1 and therefore make the equation true.
Now I have found four real numbers that satisfy the equation.
The problem asks me to determine the sum of all such numbers.
I can add them up and simplify.
The sum is $\frac{5+\sqrt{17}}{2}+\frac{5-\sqrt{17}}{2}+2+\sqrt{3}+2-\sqrt{3}$.
I can combine the fractions with the same denominator and cancel out the square roots that have opposite signs.
I get $\frac{10}{2}+4=9$.
This is the final answer.