Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) See Example 9. 9x^2 + 11x + 4 = 0

The discriminant is a key component of a quadratic equation in the form ax^2 + bx + c = 0, represented by the formula D = b^2 - 4ac. It helps determine the nature of the roots of the equation. If D > 0, there are two distinct real solutions; if D = 0, there is exactly one real solution; and if D < 0, the solutions are nonreal complex numbers.

The nature of solutions refers to the characteristics of the roots of a quadratic equation based on the value of the discriminant. Distinct solutions can be rational (if they can be expressed as a fraction of integers), irrational (if they cannot be expressed as such but are still real), or nonreal complex (if the solutions involve imaginary numbers). Understanding this helps in predicting the behavior of the quadratic function.

Quadratic equations are polynomial equations of degree two, typically expressed in the standard form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. They can be solved using various methods, including factoring, completing the square, or applying the quadratic formula. The study of their solutions and properties is fundamental in algebra and has applications in various fields.