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:

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

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

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Question: If $ x + \frac{1}{x} = 4 $, what is the value of $ x^2 + \frac{1}{x^2} $?
We square both sides of the given equation:
\left(x + \frac{1}{x}\right)^2 = 4^2 \Rightarrow x^2 + 2 + \frac{1}{x^2} = 16

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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

  1. Simplify constants: Move all non-squared terms to the right.
  2. Factor coefficients of squared terms: Normalize both squared terms to 1 by dividing by the constant on the right.
  3. 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: