if a `lt c lt b, ` then check the nature of roots of the equation
`(a -b)^(2) x^(2) + 2(a+ b - 2c)x + 1 = 0`

Explanation

To determine the nature of the roots of the quadratic equation

\[ (a - b)^2 x^2 + 2(a + b - 2c)x + 1 = 0, \]

we will follow these steps:

### Step 1: Identify the coefficients
In a quadratic equation of the form \(Ax^2 + Bx + C = 0\), we identify:
- \(A = (a - b)^2\)
- \(B = 2(a + b - 2c)\)
- \(C = 1\)

### Step 2: Calculate the discriminant
The nature of the roots is determined by the discriminant \(D\), given by:

\[ D = B^2 - 4AC. \]

Substituting the identified coefficients:

\[ D = [2(a + b - 2c)]^2 - 4[(a - b)^2 \cdot 1]. \]

### Step 3: Simplify the discriminant
Calculating \(D\):

\[ D = 4(a + b - 2c)^2 - 4(a - b)^2. \]

Factoring out the common term:

\[ D = 4\left[(a + b - 2c)^2 - (a - b)^2\right]. \]

### Step 4: Use the difference of squares
We can use the difference of squares formula:

\[ x^2 - y^2 = (x - y)(x + y). \]

Let \(x = a + b - 2c\) and \(y = a - b\):

\[ D = 4\left[(a + b - 2c - (a - b))(a + b - 2c + (a - b))\right]. \]

This simplifies to:

\[ D = 4\left[(b - 2c + b)(2a - 2c)\right] = 4\left[(2b - 2c)(2a - 2c)\right]. \]

### Step 5: Factor out the common terms
Factoring out \(4\):

\[ D = 8(b - c)(a - c). \]

### Step 6: Analyze the discriminant
Given that \(a < c < b\), both \(b - c\) and \(a - c\) are negative. Therefore, the product \(D = 8(b - c)(a - c)\) will be positive because the product of two negative numbers is positive.

### Conclusion
Since \(D > 0\), the roots of the quadratic equation are real and distinct.

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