The intersection of X-axis in parabola primarily depends on the value of

Options

A. D
B. An irrational
C. b
D. none of these

Answer

To determine the intersection of the X-axis with a parabola, we need to analyze the quadratic equation of the parabola, which is typically given in the form:

\[ ax^2 + bx + c = 0 \]

### Step-by-Step Solution:

1. **Understanding the Quadratic Equation**:
The equation of a parabola can be represented as a quadratic equation. The general form is \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants.

**Hint**: Recall that the intersection points with the X-axis occur where \( y = 0 \).

2. **Finding the Roots**:
The roots of the quadratic equation (where the parabola intersects the X-axis) can be found using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{D}}{2a} \]

where \( D \) (the discriminant) is given by:

\[ D = b^2 - 4ac \]

**Hint**: The discriminant \( D \) helps us determine the nature of the roots.

3. **Analyzing the Discriminant**:
The value of the discriminant \( D \) tells us how many times the parabola intersects the X-axis:
- If \( D > 0 \): There are two distinct real roots, meaning the parabola intersects the X-axis at two points.
- If \( D = 0 \): There is exactly one real root (the parabola touches the X-axis at one point).
- If \( D < 0 \): There are no real roots, meaning the parabola does not intersect the X-axis at all.

**Hint**: Check the sign of the discriminant to determine the number of intersections.

4. **Conclusion**:
Therefore, the intersection of the X-axis with the parabola primarily depends on the value of the discriminant \( D \).

**Final Answer**: The intersection of the X-axis in a parabola primarily depends on the value of the discriminant \( D \).