When the parabola never intersects the X-axis then which of the following is certainly true?

Options

A. `Dgt0`
B. `alt0`
C. D = 0
D. `b^2le4ac`

Answer

To determine when a parabola never intersects the X-axis, we need to analyze the conditions under which a quadratic equation has no real roots. The general form of a quadratic equation is:

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

The discriminant \( D \) of this quadratic equation is given by:

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

The nature of the roots of the quadratic equation depends on the value of the discriminant \( D \):

1. If \( D > 0 \), the parabola intersects the X-axis at two distinct points (two real roots).
2. If \( D = 0 \), the parabola touches the X-axis at one point (one real root).
3. If \( D < 0 \), the parabola does not intersect the X-axis at all (no real roots).

Since the question asks for the condition when the parabola never intersects the X-axis, we focus on the case where \( D < 0 \).

Therefore, we set up the inequality:

\[ b^2 - 4ac < 0 \]

Rearranging this inequality gives us:

\[ b^2 < 4ac \]

This means that for the parabola to never intersect the X-axis, the condition that must be satisfied is:

\[ b^2 < 4ac \]

Thus, the correct answer is that the condition \( b^2 < 4ac \) must hold true.

### Final Answer:
The condition that must be certainly true when the parabola never intersects the X-axis is:

\[ b^2 < 4ac \]