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

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 Context

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:

Key Insights

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

🔗 Related Articles You Might Like:

📰 This Simple Step Reveals Hidden Oil Problems – No Mechanic Needed 📰 How Experts Check Engine Oil – The Shocking Truth No One Tells You 📰 You Won’t Believe What Secret White Sauce Does to Your Oven Ribs 📰 From Scientific Precision To Everyday Power Newton Meters Exploded As Foot Pounds 📰 From Scoreboard Shock To Commentary Gold The Northwestern Vs Badgers Battle 📰 From Screen To Heart The Complete Natasha Richardson Movie Collection You Cant Miss 📰 From Screen To Stage Nacho Libres Iconic Look Transformed Into Iconic Costume 📰 From Secret Whispers To Stolen Gigglesneebabys Voice Is The Real Mystery Behind Baby Magic 📰 From Shadows To Spotlight Myrna Colley Lees Emotional Journey Exposes The Truth 📰 From Shockwaves To Scandals How Nude Color Changed Everything You Thought About Art 📰 From Silence To Fire Nag Champa Shocks Everyone With Her True Emotions 📰 From Silence To Fire The Untold Story Of Nautica Malone 📰 From Silence To Power The Hidden Truth About What Par Really Does 📰 From Silence To Power The Shocking Truth Behind Ma Lvarezs Wrath 📰 From Silence To Scandal Peter Hernandezs Hidden Moment Take Over 📰 From Silence To Shock Packwoods Reveals What Could Never Be Forgotten 📰 From Silence To Stardom The Raw Power Behind My Bassetts Perfect Wavewatch It Now 📰 From Silent Screams To Obsession The Hidden Truth Of My Licons

Final Thoughts

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: