If the roots of the equation `ax ^(2) + bx + c =0` are real and distinct, then

Options

A. both roots are greater than `- (b)/(2a)`
B. both roots are greater than`- (b)/(2a)`
C. one of the roots exceeds − `(b)/(2a)`
D. None of the above

Answer

To determine the condition under which the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are real and distinct, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Quadratic Equation**:
The general form of a quadratic equation is given by:
\[ ax^2 + bx + c = 0 \]

2. **Understand the Discriminant**:
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
where \( D \) is the discriminant, defined as:
\[ D = b^2 - 4ac \]

3. **Condition for Real and Distinct Roots**:
For the roots to be real and distinct, the discriminant must be greater than zero:
\[ D > 0 \]
This means:
\[ b^2 - 4ac > 0 \]

4. **Conclusion**:
Therefore, the condition for the roots of the quadratic equation \( ax^2 + bx + c = 0 \) to be real and distinct is:
\[ b^2 - 4ac > 0 \]