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. 3x^2 + 5x + 2 = 0

Key Concepts

Discriminant

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 (a repeated root); and if D < 0, the solutions are nonreal complex numbers.

Types of Solutions

The solutions of a quadratic equation can be classified based on the value of the discriminant. Real solutions occur when the discriminant is non-negative, while nonreal complex solutions arise when the discriminant is negative. Additionally, real solutions can be further categorized as rational or irrational, depending on whether they can be expressed as fractions of integers or not.

Quadratic Formula

The quadratic formula, x = (-b ± √D) / (2a), provides a method for finding the roots of a quadratic equation. While the question specifies not to solve the equation, understanding this formula is essential for interpreting the discriminant's implications on the nature of the solutions. The formula shows how the discriminant directly influences whether the solutions are real or complex.