If a and b are the different real roots of the equations `px^(2) + qx + r = 0` (where p.q.r are positive) then which of the following is correct ?

Options

A. `a gt 0. b gt 0`
B. `a lt 0. b lt 0`
C. `a gt 0. b lt 0`
D. `a lt 0. b gt 0`

Answer

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 tasked with determining the nature of the roots \( a \) and \( b \) of this equation.

### Step-by-Step Solution:

1. **Identify the Quadratic Equation**:
The equation is \( px^2 + qx + r = 0 \).

2. **Determine the Discriminant**:
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula:
\[ D = b^2 - 4ac \]
For our equation, \( a = p \), \( b = q \), and \( c = r \). Therefore, the discriminant becomes:
\[ D = q^2 - 4pr \]

3. **Condition for Real Roots**:
For the roots to be real and different, the discriminant must be positive:
\[ D > 0 \implies q^2 - 4pr > 0 \]

4. **Roots of the Quadratic Equation**:
The roots \( a \) and \( b \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting our values, we have:
\[ x = \frac{-q \pm \sqrt{q^2 - 4pr}}{2p} \]

5. **Analyzing the Roots**:
Since \( p, q, r \) are all positive, the term \( -q \) will be negative. The term \( \sqrt{q^2 - 4pr} \) will also be a positive value (since \( D > 0 \)). Thus:
- The first root \( a = \frac{-q + \sqrt{q^2 - 4pr}}{2p} \) will be negative because \( -q \) is negative and \( \sqrt{q^2 - 4pr} \) is less than \( q \) (since \( D > 0 \)).
- The second root \( b = \frac{-q - \sqrt{q^2 - 4pr}}{2p} \) will also be negative, as it is the sum of a negative number and a larger negative number.

6. **Conclusion**:
Since both roots \( a \) and \( b \) are negative, we conclude that:
\[ a < 0 \quad \text{and} \quad b < 0 \]

### Final Answer:
The correct statement is that both roots \( a \) and \( b \) are negative.

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