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 Hidden Gem of QBert Will Change How You Understand AI Text Models Forever! 📰 QBert Just Answered Everything—Build Your Own Chatbot with This Revolutionary Tool! 📰 The Untold Secret Behind QBert That Experts Are Finally Getting! 📰 This Drink Claims To Transform Your Energy In Momentsfeel The Magic Now 📰 This Dusty Italian Town Holds The Shocking Reason Flo Changed Forever 📰 This Eerie Mask Doesnt Hide Facesit Drains Souls 📰 This Elf Wont Just Reposition Toysits Secretly Filming Your Reactions 📰 This Emotional Turmoil Isnt Just Feelings Its A Figurative Cry You Cant Ignore 📰 This Episode Will Shatter Everything You Know About Tech Chat 📰 This Essay Reveals The Dark Truth Lurking In Club Culture 📰 This Excessively Beautiful German Shepherd Mix Shocks Everyone With Golden Retriever Glow 📰 This Explosive Force Turns Dark Nights Into Legendsnever Look Away 📰 This Explosive Rash Steps Outside The Norm Is It Just A Skin Glitch 📰 This Explosive Spell Could Be Your Last Chanceor Your Last Disaster 📰 This Extreme Gravity Run Will Leave You Breathless And Begging More 📰 This Fake Gruyere Cheese Is So Close Youll Forget The Difference 📰 This False Leather Trick Is Mounting Injuries And Ruining Convs 📰 This Fathers Day Moment Shattered Every Expectation You Imagine

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: