Final answer:
The discriminant in the quadratic formula determines the number and type of roots of a quadratic equation. It has four cases: positive for two real roots, zero for one real root, and negative for two complex roots. For a quadratic equation with a positive discriminant, there are two distinct real roots.
Step-by-step explanation:
The four cases of the discriminant in the quadratic formula relate to the nature of the roots of a quadratic equation of the form ax² + bx + c = 0. The discriminant is part of the quadratic formula under the square root, b² - 4ac, and it determines the number and type of roots the equation has:
If the discriminant is positive, the equation has two distinct real roots.
If the discriminant is zero, the equation has exactly one real root or a repeated root.
If the discriminant is negative, the equation has two complex roots, which are conjugates of each other.
For the quadratic equation with constants a = 4.90, b = 14.3, and c = - 20.0, you would calculate the discriminant as follows:
Discriminant = (14.3)² - 4(4.90)(-20.0) = 204.49 + 392 = 596.49
Since the discriminant is positive, this particular quadratic equation has two distinct real roots.