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:

📰 Seed ZZZ: The Secret Revolution Changing How We Sleep Forever! 📰 Zzz Seed Hacks: Transform Your Nights in Seconds—Shocking Results Inside! 📰 Mind-Blowing Benefits of Seed Zzz You’ve Never Heard of Before! 📰 You Havent Laughed Until You Saw This Sus Dog Memewatch Until You Cry 📰 You Hit A Brick Wallheres What You Must Do Next Inside The Silent Whisper Of Dead Ends 📰 You Need Suikoden Ii To Unlock Gameplay Deep Like Never Beforestart Playing Today 📰 You Need The Stussy One Piece Artworkits The Ultimate Combination For Style Fans 📰 You Need This Swim Dress Shorts Duoheres Why Its Disrupting Beach Mode 📰 You Need This Tears Of The Kingdom Walkthrough See The Emotion That Will Break Your Heart 📰 You Need To Watch This How Super Mario Switch Transforms Gameplay Like Magic 📰 You Never Dreamed Of This In Super Mario 3D World Super Marioshocking New Gameplay 📰 You Never Knew Super Hero Movies Could Undelete Your Soul 0 📰 You Never Knew Tejon Outlets Ca Sold Outheres What Really Happened 📰 You Never Saw Gremlins In A Payment Pipelinestripe Gremlins Take Over 📰 You Never Saw This Virus Its Invading Cities Like A Wildfire Shocking Details Inside 📰 You Never Saw Toy Story 3 Comingthe Shocking Truth Behind The Heartbreaking Ending 📰 You Said Ittekken 8 Is Powering Up On Ps5 With Unbelievable Graphics And Moves 📰 You Still Cant Guess What Stranger Things Season 3 Is Actually Aboutwe Revealed The Shocking Twist

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: