In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation. 2x^2 - 11x + 3 = 0

Key Concepts

Discriminant

The discriminant is a key component of the quadratic formula, given by the expression b² - 4ac for a quadratic equation in the form ax² + bx + c = 0. It helps determine the nature of the roots of the equation. If the discriminant is positive, there are two distinct real solutions; if it is zero, there is exactly one real solution; and if it is negative, there are two complex solutions.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, typically expressed as ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to this equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for analyzing their solutions.

Types of Solutions

The types of solutions for a quadratic equation are categorized based on the value of the discriminant. Real solutions occur when the discriminant is non-negative, while complex solutions arise when it is negative. This classification is crucial for understanding the behavior of the quadratic function and its graph, which can intersect the x-axis at different points depending on the nature of the solutions.