The number of roots a quadratic equation has can be determined by examining its discriminant ($\Delta$) from the quadratic formula:

$\Delta = b^2 - 4ac$

Here’s the interpretation of the discriminant:

1. If $\Delta > 0$:
   • The quadratic equation has two distinct real roots.
2. If $\Delta = 0$:
   • The quadratic equation has exactly one real root (also called a repeated or double root).
3. If $\Delta < 0$:
   • The quadratic equation has no real roots, but it has two complex roots (conjugates).

Quadratic Formula

The roots of the quadratic equation $ax^2 + bx + c = 0$ are given by:

$x = \frac{-b \pm \sqrt{\Delta}}{2a}$

If the discriminant is negative, the square root introduces imaginary components, resulting in complex roots.

Summary:

• Calculate $b^2 - 4ac$.
• Interpret $\Delta$ as above to determine the number and type of roots.

---

Would you like an example of solving for the roots or further clarification? Here are 5 related questions to expand on this:

1. How do you solve a quadratic equation with two real roots?
2. What is the significance of the coefficients $a$, $b$, and $c$ in the quadratic formula?
3. Can a quadratic equation ever have three roots?
4. How do complex roots of a quadratic equation relate to the graph of the parabola?
5. What happens when $a = 0$ in the equation $ax^2 + bx + c = 0$?

Tip: The discriminant not only tells the number of roots but also indicates the nature of the graph of the quadratic function ($f(x)$). A positive discriminant suggests the parabola intersects the x-axis twice, zero indicates it touches once, and a negative discriminant means it doesn't touch the x-axis.