## Solving the Quadratic Equation (m-1)x² - 2(m-1)x + 1 = 0

This article will explore the quadratic equation **(m-1)x² - 2(m-1)x + 1 = 0**, focusing on finding its solutions and understanding the impact of the parameter 'm'.

### Understanding the Equation

The equation is a quadratic equation in the variable 'x', with the coefficient of the quadratic term being (m-1), the coefficient of the linear term being -2(m-1), and the constant term being 1. The parameter 'm' affects the coefficients, thereby influencing the solutions.

### Finding the Solutions

We can solve for 'x' using the quadratic formula:

**x = (-b ± √(b² - 4ac)) / 2a**

Where:

- a = (m-1)
- b = -2(m-1)
- c = 1

Substituting these values into the quadratic formula, we get:

**x = (2(m-1) ± √((-2(m-1))² - 4(m-1)(1))) / 2(m-1)**

Simplifying the equation, we get:

**x = (2(m-1) ± √(4(m-1)² - 4(m-1))) / 2(m-1)**

**x = (2(m-1) ± √(4(m-1)(m-2))) / 2(m-1)**

**x = (2(m-1) ± 2√((m-1)(m-2))) / 2(m-1)**

**x = (m-1 ± √((m-1)(m-2))) / (m-1)**

This gives us two solutions for 'x':

**x₁ = (m-1 + √((m-1)(m-2))) / (m-1)**

**x₂ = (m-1 - √((m-1)(m-2))) / (m-1)**

### Analyzing the Solutions

The solutions of the equation depend on the value of the parameter 'm'. Let's analyze the different scenarios:

**1. m = 1:**

When m = 1, the equation becomes:

**(1-1)x² - 2(1-1)x + 1 = 0**

This simplifies to:

**0x² + 0x + 1 = 0**

This equation has no real solutions as the coefficient of the quadratic term is zero, and the constant term is non-zero.

**2. m = 2:**

When m = 2, the equation becomes:

**(2-1)x² - 2(2-1)x + 1 = 0**

This simplifies to:

**x² - 2x + 1 = 0**

This equation can be factored as:

**(x-1)² = 0**

Therefore, the equation has one real solution:

**x = 1**

**3. m ≠ 1 and m ≠ 2:**

When m is neither 1 nor 2, the equation has two real solutions, as determined by the quadratic formula.

### Conclusion

The quadratic equation **(m-1)x² - 2(m-1)x + 1 = 0** provides a diverse range of solutions depending on the value of the parameter 'm'. It demonstrates how a seemingly simple equation can exhibit complex behavior, emphasizing the importance of analyzing the impact of coefficients and parameters in solving mathematical problems.