One of the root of the quadratic equations `ax^(2) + bx + c =0 a !=0` is positive and the other is negative. The condition for this to happen is :

Options

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

Answer

To determine the condition under which one root of the quadratic equation \( ax^2 + bx + c = 0 \) (where \( a \neq 0 \)) is positive and the other is negative, we can follow these steps:

### Step 1: Understand the nature of the roots
For a quadratic equation, the roots can be determined using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
To have one positive root and one negative root, the roots must be distinct and of opposite signs.

### Step 2: Condition for distinct roots
The condition for the roots to be distinct is given by the discriminant:
\[ D = b^2 - 4ac > 0 \]
This ensures that the roots are real and different.

### Step 3: Condition for one positive and one negative root
For one root to be positive and the other to be negative, we need to analyze the sum and product of the roots.

1. **Sum of the roots**: The sum of the roots \( r_1 + r_2 \) is given by:
\[ r_1 + r_2 = -\frac{b}{a} \]
For one root to be positive and the other negative, the sum must be positive. Therefore:
\[ -\frac{b}{a} > 0 \implies b < 0 \quad (\text{since } a > 0) \]

2. **Product of the roots**: The product of the roots \( r_1 \cdot r_2 \) is given by:
\[ r_1 \cdot r_2 = \frac{c}{a} \]
For one root to be positive and the other negative, the product must be negative:
\[ \frac{c}{a} < 0 \implies c < 0 \quad (\text{since } a > 0) \]

### Step 4: Conclusion
From the above conditions, we conclude:
- \( b < 0 \) (the coefficient of \( x \) must be negative)
- \( c < 0 \) (the constant term must be negative)

Thus, the condition for one root to be positive and the other to be negative in the quadratic equation \( ax^2 + bx + c = 0 \) is:
\[ b < 0 \quad \text{and} \quad c < 0 \]