In Exercises 49–56, identify each equation without completing the square. 4x^2 + 4y^2 + 12x + 4y + 1 = 0

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically in the form ax^2 + bx + c = 0. In the context of the given equation, it involves both x and y variables, indicating a conic section. Understanding the structure of quadratic equations is essential for identifying their types and properties.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The main types include circles, ellipses, parabolas, and hyperbolas. The given equation can represent a conic section, and recognizing its form helps in determining which type it is without completing the square.

Standard Form of Conic Sections

The standard form of conic sections provides a way to express equations in a recognizable format, such as (x-h)^2/a^2 + (y-k)^2/b^2 = 1 for circles and ellipses. Identifying the standard form allows for easier classification and analysis of the conic represented by the equation, which is crucial for solving the problem at hand.