If p,q,r are rational numbers. Then the root of the equation `x^(2) - 2px + p^(2) - q^(2) + 2qr - r^(2) = 0` are :

Options

A. Complex
B. Pure imaginary
C. Irrational
D. Rational

Answer

To solve the quadratic equation \( x^2 - 2px + p^2 - q^2 + 2qr - r^2 = 0 \) and determine the nature of its roots, we will follow these steps:

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

### Step 2: Calculate the discriminant
The discriminant \( D \) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-2p)^2 - 4 \cdot 1 \cdot (p^2 - q^2 + 2qr - r^2) \]

### Step 3: Simplify the discriminant
Calculating \( D \):
\[ D = 4p^2 - 4(p^2 - q^2 + 2qr - r^2) \]
Distributing the \( -4 \):
\[ D = 4p^2 - 4p^2 + 4q^2 - 8qr + 4r^2 \]
This simplifies to:
\[ D = 4q^2 - 8qr + 4r^2 \]

### Step 4: Factor the discriminant
We can factor \( D \):
\[ D = 4(q^2 - 2qr + r^2) \]
Recognizing that \( q^2 - 2qr + r^2 \) can be expressed as:
\[ D = 4(q - r)^2 \]

### Step 5: Analyze the discriminant
Since \( q \) and \( r \) are rational numbers, \( (q - r)^2 \) is always non-negative. Therefore, \( D \) is non-negative:
\[ D \geq 0 \]
This means the roots of the quadratic equation are real.

### Step 6: Determine the nature of the roots
Since \( D \) is a perfect square (specifically \( 4(q - r)^2 \)), the roots are not only real but also rational.

### Conclusion
Thus, the roots of the equation \( x^2 - 2px + p^2 - q^2 + 2qr - r^2 = 0 \) are **rational**.

### Final Answer
The roots are rational numbers.

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