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

Verified step by step guidance

1. Identify the general form of the conic section equation: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).
2. Compare the given equation \$4x^2 + 4y^2 + 12x + 4y + 1 = 0\( with the general form to identify coefficients: \)A = 4\(, \)B = 0\(, \)C = 4\(, \)D = 12\(, \)E = 4\(, \)F = 1$.
3. Notice that \(A = C\) and \(B = 0\), which suggests the equation is a circle.
4. To confirm, check if the equation can be rewritten in the form \((x - h)^2 + (y - k)^2 = r^2\) by completing the square for both \(x\) and \(y\) terms.
5. Since the coefficients of \(x^2\) and \(y^2\) are equal and there is no \(xy\) term, the equation represents a circle.

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.