The nature of the roots of the quadratic equation $ax^2 + bx + c = 0$ is determined by the discriminant, denoted as $\Delta$, and given by the formula:

$\Delta = b^2 - 4ac$

1. Two Distinct Real Roots:

The quadratic equation has two distinct real roots if the discriminant is positive ($\Delta > 0$). In this case, the roots are real and unequal, and they can be found using the quadratic formula:

$x_1 = \frac{-b + \sqrt{\Delta}}{2a}, \quad x_2 = \frac{-b - \sqrt{\Delta}}{2a}$

2. One Real Root (Repeated Root):

The quadratic equation has one real root, also known as a repeated root or double root, if the discriminant is zero ($\Delta = 0$). In this case, both roots are equal, and the solution is:

$x = \frac{-b}{2a}$

3. No Real Roots:

The quadratic equation has no real roots if the discriminant is negative ($\Delta < 0$). In this case, the roots are complex and conjugate to each other. The roots are expressed as:

$x_1 = \frac{-b + i\sqrt{|\Delta|}}{2a}, \quad x_2 = \frac{-b - i\sqrt{|\Delta|}}{2a}$

Here, $i$ represents the imaginary unit, $i = \sqrt{-1}$.

Summary of Conditions:

• Two distinct real roots if $\Delta > 0$
• One real root (repeated root) if $\Delta = 0$
• No real roots (complex roots) if $\Delta < 0$

Would you like more details or have any questions?

Related Questions:

1. How is the discriminant related to the graph of a quadratic function?
2. How can you determine the axis of symmetry for a quadratic equation?
3. What happens to the roots when $a$, $b$, or $c$ are changed?
4. How does the quadratic formula derive from completing the square?
5. What is the geometric interpretation of a repeated root?

Tip: Always check the discriminant first when solving quadratic equations—it quickly tells you the nature of the roots!