If p and q are positive integers. Then which one of the following equatins has `p - sqrt(q)` as one of its roots ?

Options

A. `x^(2) - 2px - (p^(2) - q) = 0`
B. `x^(2) - 2px + (p^(2) - q) = 0`
C. `x^(2) + 2px - (p^(2) - q) = 0`
D. `x^(2) + 2px + (p^(2) - q) = 0`

Answer

To determine which equation has \( p - \sqrt{q} \) as one of its roots, we will substitute \( x = p - \sqrt{q} \) into each of the given equations and check if the equation holds true (i.e., equals zero).

### Step-by-Step Solution:

1. **Identify the equations**: We have the following equations to check:
- (A) \( x^2 - 2px - (p^2 - q) = 0 \)
- (B) \( x^2 - 2px + (p^2 - q) = 0 \)
- (C) \( x^2 + 2px - (p^2 - q) = 0 \)
- (D) \( x^2 + 2px + (p^2 - q) = 0 \)

2. **Substituting into equation (A)**:
\[ x = p - \sqrt{q} \]
Substitute into (A):
\[ (p - \sqrt{q})^2 - 2p(p - \sqrt{q}) - (p^2 - q) = 0 \]
Expanding:
\[ p^2 - 2p\sqrt{q} + q - 2p^2 + 2p\sqrt{q} - p^2 + q = 0 \]
Simplifying:
\[ 2q - 2p^2 = 0 \implies q = p^2 \quad \text{(not necessarily true for all positive integers)} \]
Thus, (A) does not hold.

3. **Substituting into equation (B)**:
Substitute into (B):
\[ (p - \sqrt{q})^2 - 2p(p - \sqrt{q}) + (p^2 - q) = 0 \]
Expanding:
\[ p^2 - 2p\sqrt{q} + q - 2p^2 + 2p\sqrt{q} + p^2 - q = 0 \]
Simplifying:
\[ 0 = 0 \quad \text{(holds true)} \]
Thus, (B) is a valid equation.

4. **Substituting into equation (C)**:
Substitute into (C):
\[ (p - \sqrt{q})^2 + 2p(p - \sqrt{q}) - (p^2 - q) = 0 \]
Expanding:
\[ p^2 - 2p\sqrt{q} + q + 2p^2 - 2p\sqrt{q} - p^2 + q = 0 \]
Simplifying:
\[ 2p^2 - 4p\sqrt{q} + 2q = 0 \quad \text{(not necessarily true for all positive integers)} \]
Thus, (C) does not hold.

5. **Substituting into equation (D)**:
Substitute into (D):
\[ (p - \sqrt{q})^2 + 2p(p - \sqrt{q}) + (p^2 - q) = 0 \]
Expanding:
\[ p^2 - 2p\sqrt{q} + q + 2p^2 - 2p\sqrt{q} + p^2 - q = 0 \]
Simplifying:
\[ 4p^2 - 4p\sqrt{q} = 0 \quad \text{(not necessarily true for all positive integers)} \]
Thus, (D) does not hold.

### Conclusion:
The only equation that holds true when substituting \( p - \sqrt{q} \) is equation (B):
\[ x^2 - 2px + (p^2 - q) = 0 \]