The equation `Px^(2) + qx + r = 0 ` (where p, q, r, all are positive ) has distinct real roots a and b .

Options

A. `a gt 0, b gt 0`
B. ` a lt 0, b lt0`
C. ` a gt 0, blt 0 `
D. ` a lt 0, b gt0`

Explanation

To solve the problem, we need to analyze the quadratic equation given by \( Px^2 + Qx + R = 0 \) where \( P, Q, R \) are all positive constants. We are asked to determine the nature of the roots of this equation.

### Step-by-Step Solution:

1. **Identify the Form of the Quadratic Equation**:
The given equation is in the standard form of a quadratic equation, which is \( ax^2 + bx + c = 0 \). Here, we have \( a = P \), \( b = Q \), and \( c = R \).

2. **Condition for Distinct Real Roots**:
For a quadratic equation to have distinct real roots, the discriminant must be positive. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Substituting our values, we have:
\[ D = Q^2 - 4PR \]
Since \( P, Q, R > 0 \), we need \( D > 0 \) for the roots to be distinct.

3. **Analyzing the Discriminant**:
To ensure that the roots are distinct, we require:
\[ Q^2 - 4PR > 0 \]
This means that \( Q^2 \) must be greater than \( 4PR \).

4. **Finding the Nature of the Roots**:
Since \( P, Q, R \) are all positive, and we have established that the discriminant is positive, we can conclude that the roots \( a \) and \( b \) are real and distinct.

5. **Sign of the Roots**:
Given that \( P, Q, R > 0 \), we can analyze the sign of the roots using Vieta's formulas. According to Vieta's, the sum of the roots \( a + b = -\frac{Q}{P} \) and the product of the roots \( ab = \frac{R}{P} \).
- Since \( Q > 0 \), it follows that \( a + b < 0 \), meaning the sum of the roots is negative.
- Since \( R > 0 \), it follows that \( ab > 0 \), meaning the product of the roots is positive.

From these two conditions, we can conclude that both roots \( a \) and \( b \) must be negative. Therefore, we have:
\[ a < 0 \quad \text{and} \quad b < 0 \]

### Final Conclusion:
The equation \( Px^2 + Qx + R = 0 \) has distinct real roots \( a \) and \( b \), both of which are negative.