4[(x - 4)^2 - 16] - 9[(y - 3)^2 - 9] = -115 \implies 4(x - 4)^2 - 64 - 9(y - 3)^2 + 81 = -115.

["Understanding the Equation 4[(x - 4)² – 16] – 9[(y – 3)² – 9] = –115: A Step-by-Step Breakdown", "Solving algebraic equations can often feel complicated at first glance—especially when parentheses and squared terms dominate. However, expanding and simplifying expressions like 4[(x – 4)² – 16] – 9[(y – 3)² – 9] = –115 reveals its hidden structure and potential applications in geometry or coordinate transformations. This article demystifies the equation and explains how to approach and interpret it.", "---", "### The Equation at a Glance", "We begin with:\n$$\n4[(x - 4)^2 - 16] - 9[(y - 3)^2 - 9] = -115\n$$", "At first, expanding this expression might seem tedious, but grouping terms carefully unlocks clarity.", "---", "### Step 1: Distribute the Constants", "Apply the distributive property:", "$$\n4(x - 4)^2 - 4 \cdot 16 - 9(y - 3)^2 + 9 \cdot 9 = -115\n$$\n$$\n4(x - 4)^2 - 64 - 9(y - 3)^2 + 81 = -115\n$$", "---", "### Step 2: Combine Constant Terms", "Group constants on the left-hand side:\n$$\n(-64 + 81) + 4(x - 4)^2 - 9(y - 3)^2 = -115\n$$\n$$\n17 + 4(x - 4)^2 - 9(y - 3)^2 = -115\n$$", "Subtract 17 from both sides:\n$$\n4(x - 4)^2 - 9(y - 3)^2 = -115 - 17 = -132\n$$", "Wait—this does not match the target simplification. Let’s double-check our algebra and approach.", "---", "### Revisiting the Expansion: Correct Simplification", "Begin again with:\n$$\n4[(x - 4)^2 - 16] - 9[(y - 3)^2 - 9] = -115\n$$", "Distribute:\n$$\n4(x - 4)^2 - 64 - 9(y - 3)^2 + 81 = -115\n$$", "Combine constants:\n$$\n4(x - 4)^2 - 9(y - 3)^2 + 17 = -115\n$$", "Move constant to right-hand side:\n$$\n4(x - 4)^2 - 9(y - 3)^2 = -115 - 17 = -132\n$$", "So:\n$$\n4(x - 4)^2 - 9(y - 3)^2 = -132\n$$", "Divide the entire equation by 132 to normalize (optional for standard form):\n$$\n\frac{(x - 4)^2}{-33} + \frac{(y - 3)^2}{\frac{132}{9}} = 1\n\quad \ ext{(Not standard ellipse form, so keep as is.)}\n$$", "Instead, observe that we can express the original form as:\n$$\n4(x - 4)^2 - 9(y - 3)^2 = -132\n$$", "Multiply both sides by –1:\n$$\n-4(x - 4)^2 + 9(y - 3)^2 = 132\n$$", "Rewriting:\n$$\n9(y - 3)^2 - 4(x - 4)^2 = 132\n$$", "This resembles the standard form of a hyperbola, centered at (4, 3).", "---", "### What Does This Mean?", "The original expression\n$$\n4[(x - 4)^2 - 16] - 9[(y - 3)^2 - 9] = -115\n$$\nis algebraically equivalent to:\n$$\n4(x - 4)^2 - 9(y - 3)^2 = -132\n$$", "It describes a hyperbola centered at (4, 3), with transverse axis along the y-axis due to the positive coefficient on $ (y - 3)^2 $.", "This form is useful in modeling relationships between deviations in $ x $ and $ y $, possibly related to conic sections, distance definitions, or optimization problems.", "---", "### Why This Algebra Matters", "While expanded, the expression involves quadratic terms in shifted coordinates, reflecting symmetry around (4, 3). Such forms appear often in:", "- Geometry, particularly hyperbolas and distance relations.\n- Physics, when modeling forces or fields dependent on relative positions.\n- Engineering design, where coordinate transformations align systems with natural centers.", "---", "### Final Notes on Solving This Equation", "To graph it or find key points:\n- Center: (4, 3)\n- Symmetry: Symmetric about both axes through center.\n- Asymptotes: From standard hyperbola form $ \frac{(y - 3)^2}{a^2} - \frac{(x - 4)^2}{b^2} = 1 $, asymptotes linear in $ x - 4 $, $ y - 3 $.\n- Vertex displacement: Adjusted for constants, centered instead of origin.", "---", "### Conclusion", "Rather than focusing on solving for $ x $ or $ y $ explicitly, recognizing the structure of expressions like\n$$\n4[(x - 4)^2 - 16] - 9[(y - 3)^2 - 9] = -115\n$$\nreveals a hyperbolic relationship centered at (4, 3). This algebraic insight paves the way for deeper geometric intuition and practical use in modeling curved relationships in coordinate space.", "Use these techniques next time you encounter complex expressions—simplify, group, and identify symmetry or standard forms to unlock meaning effortlessly.", "---", "Keywords for SEO:\n- Equation solving\n- Hyperbola equation\n- Coordinate geometry\n- Algebraic simplification\n- 4[(x – 4)² – 16] – 9[(y – 3)² – 9] = –115\n- Center of conic\n- Hyperbola center at (4,3)\n- Conic sections algebra", "Meta Description:\nUnlock the structure of the equation 4[(x – 4)² – 16] – 9[(y – 3)² – 9] = –115 by expanding, simplifying, and identifying its hyperbolic form centered at (4, 3) for clearer geometric interpretation."]