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

To analyze the roots of the quadratic equation \( ax^2 + bx + c = 0 \), we rely on the **discriminant**, denoted by \( \Delta \). The discriminant is derived from the coefficients of the quadratic equation and is given by:

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

### Conditions for the nature of roots:

1. **Two Distinct Real Roots**:  
   The quadratic equation has two distinct real roots if the discriminant is **positive**.
   \[
   \Delta > 0 \quad \text{(i.e., \( b^2 - 4ac > 0 \))}
   \]
   In this case, the roots are given by 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 exactly **one real root** if the discriminant is **zero**.
   \[
   \Delta = 0 \quad \text{(i.e., \( b^2 - 4ac = 0 \))}
   \]
   In this case, the quadratic has a repeated root, and the solution is:
   \[
   x = \frac{-b}{2a}
   \]

3. **No Real Roots (Complex Roots)**:  
   The quadratic equation has **no real roots** if the discriminant is **negative**.
   \[
   \Delta < 0 \quad \text{(i.e., \( b^2 - 4ac < 0 \))}
   \]
   In this case, the equation has two complex conjugate roots:
   \[
   x_1 = \frac{-b + i\sqrt{-\Delta}}{2a}, \quad x_2 = \frac{-b - i\sqrt{-\Delta}}{2a}
   \]
   Here, \( i \) represents the imaginary unit.

### Summary of Conditions:
- \( \Delta > 0 \): Two distinct real roots.
- \( \Delta = 0 \): One real root (repeated).
- \( \Delta < 0 \): No real roots, but two complex conjugate roots.

Would you like further details or have any questions?

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Here are 5 related questions to deepen your understanding:
1. How can you apply the discriminant to solve a specific quadratic equation?
2. What happens to the nature of roots if \( a = 0 \) in a quadratic equation?
3. How does the quadratic formula derive from completing the square method?
4. Can a quadratic equation have irrational roots? Under what conditions?
5. What is the geometric significance of a repeated real root?

**Tip:** The discriminant not only helps classify the roots but also indicates how "spread out" the roots are when they exist. A larger discriminant means the roots are further apart.

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.

Learn how the discriminant determines the nature of roots in a quadratic equation ax^2 + bx + c = 0. Explore conditions for two distinct real roots, one real root, or no real roots through detailed explanations and formulas. Suitable for Grades 9-11.

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