Sum of all real roots of equation |x–2|^2+|x–2|–2=0
The given equation is |x–2|^2 + |x–2| – 2 = 0. We need to find the sum of all real roots of this equation.

Analysis of the Equation:
- Notice that the term |x–2| can be treated as a single variable. Let's denote it as y.
- The equation now becomes y^2 + y – 2 = 0, which is a quadratic equation in y.
- Solve this quadratic equation to find the possible values of y.

Solving the Quadratic Equation:
- Factor the quadratic equation: (y + 2)(y – 1) = 0.
- Set each factor to zero: y + 2 = 0 and y – 1 = 0.
- Solve for y: y = -2 and y = 1.

Substitute back x–2 for y:
- We had y = |x–2|. So, we get |x–2| = -2 and |x–2| = 1.
- However, absolute value cannot be negative, so we discard the solution |x–2| = -2.
- Solve |x–2| = 1: x–2 = 1 or x–2 = -1.
- This gives x = 3 and x = 1 as the roots of the equation.

Sum of Real Roots:
- The sum of the real roots is 3 + 1 = 4.
Therefore, the sum of all real roots of the equation |x–2|^2 + |x–2| – 2 = 0 is 4.