To solve the equation:

$x^2 - 2x + 1 = 0$

Step 1: Factoring the equation

Notice that the quadratic expression can be factored:

$x^2 - 2x + 1 = (x - 1)^2$

Thus, the equation is equivalent to:

$(x - 1)^2 = 0$

Step 2: Solve for x

To solve $(x - 1)^2 = 0$, take the square root of both sides:

$x - 1 = 0$

So:

$x = 1$

Solution

The solution to the equation is:

$x = 1$

Would you like more details on the factoring process or have any other questions? Here are five related questions to deepen your understanding:

1. What is the general method for factoring quadratic equations?
2. How can the quadratic formula be used to solve $x^2 - 2x + 1 = 0$?
3. What does it mean when a quadratic equation has a "double root"?
4. How do you determine the number of solutions a quadratic equation has?
5. What are some real-life applications of solving quadratic equations?

Tip: If the discriminant of a quadratic equation ($b^2 - 4ac$) is zero, it indicates the equation has a double root. This means there is exactly one solution, but it's counted twice.