https://file.aiquickdraw.com/mathbot/file/1725800214219q7r892bz.jpg

This activity requires determining the **discriminant** of each quadratic equation to figure out the **nature of the roots**. The discriminant \( \Delta \) of a quadratic equation of the form \( ax^2 + bx + c = 0 \) is given by:

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

- If \( \Delta > 0 \), the equation has two distinct real roots.
- If \( \Delta = 0 \), the equation has exactly one real root (repeated).
- If \( \Delta < 0 \), the equation has two complex (non-real) roots.

Let’s solve each quadratic equation based on the image.

1. **\( x^2 + 6x + 8 = 0 \)**  
   - \( a = 1, b = 6, c = 8 \)  
   - Discriminant \( \Delta = 6^2 - 4(1)(8) = 36 - 32 = 4 \)  
   - Nature of roots: **Two distinct real roots** (since \( \Delta > 0 \)).

2. **\( x^2 + 5x + 20 = 0 \)**  
   - \( a = 1, b = 5, c = 20 \)  
   - Discriminant \( \Delta = 5^2 - 4(1)(20) = 25 - 80 = -55 \)  
   - Nature of roots: **Two complex roots** (since \( \Delta < 0 \)).

3. **\( x^2 + 9x + 20 = 0 \)**  
   - \( a = 1, b = 9, c = 20 \)  
   - Discriminant \( \Delta = 9^2 - 4(1)(20) = 81 - 80 = 1 \)  
   - Nature of roots: **Two distinct real roots** (since \( \Delta > 0 \)).

4. **\( 2x^2 + 4x + 3 = 0 \)**  
   - \( a = 2, b = 4, c = 3 \)  
   - Discriminant \( \Delta = 4^2 - 4(2)(3) = 16 - 24 = -8 \)  
   - Nature of roots: **Two complex roots** (since \( \Delta < 0 \)).

5. **\( x^2 + 2x + 1 = 0 \)**  
   - \( a = 1, b = 2, c = 1 \)  
   - Discriminant \( \Delta = 2^2 - 4(1)(1) = 4 - 4 = 0 \)  
   - Nature of roots: **One real repeated root** (since \( \Delta = 0 \)).

6. **\( 3x^2 + 7x + 2 = 0 \)**  
   - \( a = 3, b = 7, c = 2 \)  
   - Discriminant \( \Delta = 7^2 - 4(3)(2) = 49 - 24 = 25 \)  
   - Nature of roots: **Two distinct real roots** (since \( \Delta > 0 \)).

7. **\( 4x^2 + 12x + 9 = 0 \)**  
   - \( a = 4, b = 12, c = 9 \)  
   - Discriminant \( \Delta = 12^2 - 4(4)(9) = 144 - 144 = 0 \)  
   - Nature of roots: **One real repeated root** (since \( \Delta = 0 \)).

8. **\( 5x^2 + 6x + 1 = 0 \)**  
   - \( a = 5, b = 6, c = 1 \)  
   - Discriminant \( \Delta = 6^2 - 4(5)(1) = 36 - 20 = 16 \)  
   - Nature of roots: **Two distinct real roots** (since \( \Delta > 0 \)).

9. **\( 10x^2 - 3x + 1 = 0 \)**  
   - \( a = 10, b = -3, c = 1 \)  
   - Discriminant \( \Delta = (-3)^2 - 4(10)(1) = 9 - 40 = -31 \)  
   - Nature of roots: **Two complex roots** (since \( \Delta < 0 \)).

10. **\( 5x^2 - 5x + 10 = 0 \)**  
    - \( a = 5, b = -5, c = 10 \)  
    - Discriminant \( \Delta = (-5)^2 - 4(5)(10) = 25 - 200 = -175 \)  
    - Nature of roots: **Two complex roots** (since \( \Delta < 0 \)).

Let me know if you'd like further clarification on any of the steps!

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### Relative Questions:
1. What are some examples where real repeated roots occur in real-world problems?
2. How does the discriminant help in determining the type of roots without solving the quadratic?
3. Can you describe situations where quadratic equations with complex roots appear in physics?
4. What other methods can be used to solve quadratic equations besides the discriminant method?
5. What happens to the roots of a quadratic equation as the discriminant approaches zero?

**Tip**: When solving quadratic equations, always simplify the equation first to its standard form \( ax^2 + bx + c = 0 \) before calculating the discriminant.

Determine the nature of the roots of the quadratic equations by using the discriminant.

Learn how to determine the nature of the roots of quadratic equations using the discriminant formula. Solve equations such as x^2 + 6x + 8 = 0 and identify whether the roots are real or complex. This method is suitable for students in Grades 8-10.

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