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.