Given the quadratic equation 𝑎𝑥$$ ^2 $$ + 𝑏𝑥 + 𝑐 = 0, discuss the conditions under which the equation has two distinct real roots, one real root, or no real roots.

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!

Given the quadratic equation 𝑎𝑥^2 + 𝑏𝑥 + 𝑐 = 0, discuss the conditions under which the equation has two distinct real roots, one real root, or no real roots.

Explore the conditions under which a quadratic equation ax^2 + bx + c = 0 has two distinct real roots, one real root, or no real roots. This lesson explains the role of the discriminant and provides step-by-step solutions for each case. Suitable for Grades 9-12 students.

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