Identify each equation without completing the square. y² - 4x + 2y + 21 = 0

Verified step by step guidance

1. Rewrite the given equation to group the terms involving \( y \) together and isolate the \( x \) terms: \( y^2 + 2y - 4x + 21 = 0 \).
2. Move the \( x \) terms and constant to the other side to focus on the \( y \) terms: \( y^2 + 2y = 4x - 21 \).
3. Recognize that the equation is quadratic in \( y \) and linear in \( x \), which suggests it might represent a parabola that opens horizontally.
4. Recall the standard form of a parabola that opens left or right is \( (y - k)^2 = 4p(x - h) \), where \( (h, k) \) is the vertex.
5. Based on the structure and terms, identify the conic as a parabola without completing the square.

Key Concepts

Identifying Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone, including circles, ellipses, parabolas, and hyperbolas. Each conic has a standard form equation involving x and y variables. Recognizing the type of conic from its general equation is essential before further manipulation.

General Form of Conic Equations

The general form of a conic equation is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. By analyzing the coefficients, especially those of x^2 and y^2, one can determine the conic type. For example, if only one variable is squared, the conic is a parabola.

Completing the Square (Conceptual Understanding)

Completing the square is a method to rewrite quadratic expressions in a form that reveals the conic's center and shape. Although the question asks to identify the conic without completing the square, understanding this process helps in recognizing the conic type from the equation's structure.