Answer:

1. 36: Two real and distinct roots.

2. -18: Two complex (non-real) roots.

3. 0: One real root (a repeated root).

4. 196: Two real and distinct roots.

5. 2012: Two real and distinct roots.

6. -17: Two complex (non-real) roots.

7. 143: Two real and distinct roots.

8. -49: Two real and distinct roots.

9. -100: Two real and equal roots.

10. 676: Two real and distinct roots.

11. 2: Two real and distinct roots.

12. -45: Two complex (non-real) roots.

13. 1: Two real and equal roots.

14. 203: Two real and distinct roots.

15. 2025: Two real and equal roots.

Verified answer

Answer:

To determine the nature of the roots of a quadratic equation based on the value of the discriminant, we can use the following criteria:

1. If the discriminant (denoted as Δ) is greater than zero (Δ > 0), then the quadratic equation has two distinct real roots. This means that the equation crosses or intersects the x-axis at two different points.

3. If the discriminant is equal to zero (Δ = 0), then the quadratic equation has exactly one real root. This means that the equation is a perfect square and just touches the x-axis at one point.

9. If the discriminant is less than zero (Δ < 0), then the quadratic equation has two complex conjugate roots. This means that the equation does not intersect or touch the x-axis, but it has solutions in the form of complex numbers involving the imaginary unit "i."

Based on the given values:

1. The discriminant is positive (Δ = 36), so the quadratic equation has two distinct real roots.

2. The discriminant is negative (Δ = -18), so the quadratic equation has two complex conjugate roots.

3. The discriminant is zero (Δ = 0), so the quadratic equation has exactly one real root.

4. The discriminant is positive (Δ = 196), so the quadratic equation has two distinct real roots.

5. The discriminant is positive (Δ = 2012), so the quadratic equation has two distinct real roots.

6. The discriminant is negative (Δ = -17), so the quadratic equation has two complex conjugate roots.

7. The discriminant is positive (Δ = 143), so the quadratic equation has two distinct real roots.

8. The discriminant is negative (Δ = -49), so the quadratic equation has two complex conjugate roots.

9. The discriminant is negative (Δ = -100), so the quadratic equation has two complex conjugate roots.

10. The discriminant is positive (Δ = 676), so the quadratic equation has two distinct real roots.

11. The discriminant is positive (Δ = 2), so the quadratic equation has two distinct real roots.

12. The discriminant is negative (Δ = -45), so the quadratic equation has two complex conjugate roots.

13. The discriminant is positive (Δ = 1), so the quadratic equation has two distinct real roots.

14. The discriminant is positive (Δ = 203), so the quadratic equation has two distinct real roots.

15. The discriminant is positive (Δ = 2025), so the quadratic equation has two distinct real roots.

Note: I have only explained the options where the discriminant is non-zero, as that determines the nature of the roots. For the options where the discriminant is not provided, it is not possible to determine the nature of the roots.