The roots of the equation `(q - r) x^(2) + (r - p) x + (p - q) = 0` are :

Options

A. `((r - p))/((q - r)) . (1)/(2)`
B. `((p - q))/((q -r)). 1`
C. `1((q -r))/((p - r)).1`
D. `((r - p))/((p - q)).1`

Answer

To find the roots of the quadratic equation \((q - r)x^2 + (r - p)x + (p - q) = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\), where:
- \(a = q - r\)
- \(b = r - p\)
- \(c = p - q\)

### Step 2: Check if \(x = 1\) is a root
We can substitute \(x = 1\) into the equation to see if it satisfies the equation:
\[ (q - r)(1^2) + (r - p)(1) + (p - q) = 0 \]
This simplifies to:
\[ (q - r) + (r - p) + (p - q) = 0 \]
Now, combine the terms:
\[ q - r + r - p + p - q = 0 \]
Notice that \(q\) cancels with \(-q\), \(r\) cancels with \(-r\), and \(p\) cancels with \(-p\), resulting in:
\[ 0 = 0 \]
Since this is true, we conclude that \(x = 1\) is indeed a root of the equation.

### Step 3: Find the other root using the product of roots
Let the roots of the equation be \(\alpha\) and \(\beta\). We already found \(\alpha = 1\). According to Vieta's formulas, the product of the roots is given by:
\[ \alpha \cdot \beta = \frac{c}{a} \]
Substituting the values of \(c\) and \(a\):
\[ 1 \cdot \beta = \frac{p - q}{q - r} \]
Thus, we have:
\[ \beta = \frac{p - q}{q - r} \]

### Step 4: Final roots
Now we have both roots:
- \(\alpha = 1\)
- \(\beta = \frac{p - q}{q - r}\)

### Conclusion
The roots of the equation \((q - r)x^2 + (r - p)x + (p - q) = 0\) are:
- \(x = 1\)
- \(x = \frac{p - q}{q - r}\)