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To determine the number of x-intercepts of the quadratic function \( f(x) = ax^2 + bx + c \) based on the given condition \( b^2 - 4ac > 0 \), we need to analyze the discriminant (\( \Delta \)) of the quadratic equation \( ax^2 + bx + c = 0 \). The discriminant is given by:

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

The number of x-intercepts (real roots) of the quadratic function depends on the value of the discriminant:

- If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
- If \( \Delta = 0 \), the quadratic equation has exactly one real root (a repeated root).
- If \( \Delta < 0 \), the quadratic equation has no real roots.

Given \( b^2 - 4ac > 0 \), the discriminant is positive. Therefore, the quadratic function has two distinct real roots.

Thus, the correct answer is:

\[ \text{The graph has two distinct x-intercepts.} \]

Would you like more details or have any questions?

Here are 5 numbered questions for you to ask:

1. How do you find the x-intercepts of a quadratic function using the quadratic formula?
2. What is the significance of the discriminant in determining the nature of the roots of a quadratic equation?
3. Can you explain how to graph a quadratic function?
4. How do you determine the vertex of a quadratic function from its standard form?
5. What are the steps to convert a quadratic function from standard form to vertex form?

**Tip:** When analyzing the roots of a quadratic function, always check the discriminant first to understand the nature and number of the roots.

Understand how to determine the number of x-intercepts of a quadratic function using the discriminant formula. This problem explores the conditions under which a quadratic function has two distinct real roots based on the value of its discriminant. Learn the quadratic formula and its application in analyzing roots of quadratic equations.

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