9(x-2)^2 - 36 + 4(y+2)^2 - 16 = -36

February 22, 2026

Solving the Quadratic Equation: Transforming 9(x−2)² − 36 + 4(y+2)² − 16 = −36

If you're studying conic sections, algebraic manipulation, or exploring how to simplify complex equations, you may have encountered this type of equation: 9(x−2)² − 36 + 4(y+2)² − 16 = −36

This equation combines quadratic terms in both x and y, and while it may look intimidating at first, solving or analyzing it reveals powerful techniques in algebra and geometry. In this article, we will walk through the step-by-step process of simplifying this equation, exploring its structure, and understanding how it represents a geometric object. By the end, you’ll know how to manipulate such expressions and interpret their meaning.

Understanding the Equation Structure

The equation is: 9(x−2)² − 36 + 4(y+2)² − 16 = −36

We begin by combining like terms on the left-hand side:

9(x−2)² + 4(y+2)² − (36 + 16) = −36 9(x−2)² + 4(y+2)² − 52 = −36

Next, add 52 to both sides to isolate the squared terms:

9(x−2)² + 4(y+2)² = 16

Now, divide both sides by 16 to get the canonical form:

(9(x−2)²)/16 + (4(y+2)²)/16 = 1 (x−2)²/(16/9) + (y+2)²/(4) = 1

This is the standard form of an ellipse centered at (2, −2), with horizontal major axis managed by the (x−2)² term and vertical minor axis managed by the (y+2)² term.

Step-by-Step Solution Overview

Simplify constants: Move all non-squared terms to the right.
Factor coefficients of squared terms: Normalize both squared terms to 1 by dividing by the constant on the right.
Identify ellipse parameters: Extract center, major/minor axis lengths, and orientation.

Key Features of the Equation

Center: The equation (x−2)² and (y+2)² indicates the ellipse centers at (2, −2).
Major Axis: Longer axis along the x-direction because 16/9 ≈ 1.78 > 4 (Note: Correction: since 16/9 ≈ 1.78 and 4 = 16/4, actually 4 > 16/9, so the major axis is along the y-direction with length 2×√4 = 4.)
Semi-major axis length: √4 = 2
Semi-minor axis length: √(16/9) = 4/3 ≈ 1.33

Why This Equation Matters

Quadratic equations like 9(x−2)² + 4(y+2)² = 16 define ellipses in the coordinate plane. Unlike parabolas or hyperbolas, ellipses represent bounded, oval-shaped curves. This particular ellipse:

Is centered at (2, −2)
Stretches farther horizontally than vertically
Is useful in optimization, geometry modeling, and physics (e.g., describing motion paths limited within a region)

Practical Applications

Understanding how to manipulate such equations helps in:

Graphing conic sections by recognizing standard forms
Solving real-world problems involving distances and constraints (e.g., building heat zones, antenna coverage areas)
Teaching mathematical aptitude, including algebra, geometry, and coordinate transformations

Final Thoughts

Mastering the technique of simplifying 9(x−2)² − 36 + 4(y+2)² − 16 = −36 gives you immediate access to identifying an ellipse’s core geometric properties. From recognizing its center and axes to translating these into spatial reasoning, this foundational skill bridges symbolic algebra and visual geometry.

If you’re working with quadratic equations, embrace the steps: simplify constants, isolate variables, normalize denominators — and you unlock a clearer understanding of conic sections.