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:

📰 A vertical farm uses LED lights that consume 180 watts per square meter. If the farm expands from 120 to 200 square meters and operates 16 hours daily, how many kilowatt-hours are used each day? 📰 Hourly fixes form a geometric sequence: 📰 A space agriculture system uses hydroponics with nutrient solution recycled daily. It loses 15% of nutrients to evaporation each day but replenishes 60 liters fresh solution. Starting with 400 liters, how many liters remain after 3 days? 📰 Verizon In Upper Marlboro 6085962 📰 Caddy Nappy Secret Moms Are Ravingheres What Makes It Unbeatable 8661770 📰 Verizon Wireless Mt Kisco Ny 8825996 📰 Download History 8502955 📰 Difference Between 401K Ira And Roth Ira 7092561 📰 Weighted Average Formula Excel 1489233 📰 Drama Mask Hacks Your Mood Learn How To Unlock Hidden Emotions Instantly 2710038 📰 Download The Microsoft Word Flowchart Template Thats Taking Productivity By Storm 2602505 📰 How Old Is Alec On The Shriners Commercial 7343300 📰 Cracked Mac Apps 8826869 📰 Frac324 8 2338286 📰 Huge Scandal Uncovered What The Department Of Health And Human Services Is Concealing Right Now 8609906 📰 Sterling Hayden 5203326 📰 Kagome Revealed The Shocking Truth Behind Her Iconic Look 5299811 📰 What Is A 457 The Surprising Reason Thousands Are Confused About This Government Program 1475979

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: