$ 4(x^2 + 6x) - 9(y^2 - 2y) = -9 $.

["Title: Understanding the Equation $ 4(x^2 + 6x) - 9(y^2 - 2y) = -9 $: A Step-by-Step Analysis for Beginners", "---", "Introduction\nWorking with conic section equations can seem daunting at first, but equations like $ 4(x^2 + 6x) - 9(y^2 - 2y) = -9 $ follow a systematic approach when analyzed thoroughly. In this SEO-optimized article, we break down this quadratic equation in two variables, why it matters, and how to interpret and solve it—perfect for students, educators, and math enthusiasts looking to strengthen their algebra foundations.", "---", "### What is the Equation $ 4(x^2 + 6x) - 9(y^2 - 2y) = -9 $?", "This is a second-degree equation combining x and y terms, fitting the general form of a hyperbola or derived conic section when simplified. Specifically, it represents a hyperbola because it contains both a positive $ x^2 $-term and a negative $ y^2 $-term. The equation mixes linear parts ($ 6x $ and $ -2y $) with quadratic (squared) parts, making it a quadratic curve in the plane.", "---", "### Why Learn About This Equation?", "- It introduces key concepts: conic sections, algebraic simplification, and transformation of curves.\n- It helps develop problem-solving skills in standard algebraic forms.\n- Understanding such equations is vital for advanced math topics including graphing and geometry.", "---", "### Step-by-Step Breakdown of the Equation", "#### Step 1: Expand the Expression\nStart by distributing the constants inside parentheses:\n$$\n4(x^2 + 6x) - 9(y^2 - 2y) = -9\n\Rightarrow 4x^2 + 24x - 9y^2 + 18y = -9\n$$", "#### Step 2: Move Constant to the Right Side\nMove the constant $-9$ to the left to form standard equality:\n$$\n4x^2 + 24x - 9y^2 + 18y + 9 = 0\n$$", "This completes simplification. This is the general quadratic form:\n$$\nAx^2 + Bxy + Cy^2 + Dx + Ey + F = 0\n$$\nHere: $ A = 4 $, $ B = 0 $, $ C = -9 $, $ D = 24 $, $ E = 18 $, $ F = 9 $", "---", "### Recognize the Conic Section Type", "The coefficients reveal the conic type:", "- $ A = 4 $, $ C = -9 $ → opposite signs ⇒ hyperbola\n- Degree of highest powers: quadratic in $x$ and $y$ ⇒ second-degree conic", "---", "### How to Analyze and Graph This Hyperbola", "While full transformation into standard form requires completing the square, we outline the key steps:", "1. Complete the square for $ x $-terms and $ y $-terms.\n2. Rewrite the equation in standard hyperbolic form.\n3. Identify transverse axis, center, asymptotes, and vertices from the completed square.", "Example of Completing the Square (for future reference):", "For $ 4x^2 + 24x $:\nFactor: $ 4(x^2 + 6x) = 4[(x+3)^2 - 9] = 4(x+3)^2 - 36 $", "For $ -9y^2 + 18y $:\nFactor: $ -9(y^2 - 2y) = -9[(y-1)^2 - 1] = -9(y-1)^2 + 9 $", "Substitute back:\n$$\n4(x+3)^2 - 36 -9(y-1)^2 + 9 = -9\n\Rightarrow 4(x+3)^2 - 9(y-1)^2 - 27 = -9\n\Rightarrow 4(x+3)^2 - 9(y-1)^2 = 18\n$$", "Divide both sides by 18 to normalize:\n$$\n\frac{(x+3)^2}{4.5} - \frac{(y-1)^2}{2} = 1\n$$", "This confirms it is a hyperbola centered at $ (-3, 1) $, opening horizontally, with $ a^2 = 4.5 $, $ b^2 = 2 $.", "---", "### Practical Applications", "Equations like this model physical phenomena and engineering designs:\n- Trajectory paths in projectile motion combined with boundaries\n- Structural arch design in architecture\n- Signal processing and hyperbolic geometry in advanced physics", "---", "### SEO Considerations\nKeywords optimized:\n- Equation $ 4(x^2 + 6x) - 9(y^2 - 2y) = -9 $\n- Hyperbola equation explanation\n- Solve $ 4x^2 + 24x - 9y^2 + 18y = -9 $ step-by-step\n- Conic sections for beginners\n- Graphing hyperbolas algebraically\n- Algebraic transformation of quadratic expressions\n- Standard form of conic sections", "---", "### Conclusion", "Understanding $ 4(x^2 + 6x) - 9(y^2 - 2y) = -9 $ opens doors to deeper knowledge of conic sections and recursive algebraic techniques. By simplifying, completing the square, and recognizing standard forms, even complex equations become manageable. Whether you’re studying equations for coursework or curiosity, mastering this form builds essential math fluency.", "---", "Keywords: $ 4(x^2 + 6x) - 9(y^2 - 2y) = -9 $, conic sections, hyperbola, algebraic simplification, quadratic form, completing the square, conic graphing, algebra tutorial, academics, math learning", "---", "Meta Description:\nLearn step-by-step how to analyze and solve the equation $ 4(x^2 + 6x) - 9(y^2 - 2y) = -9 $. Understand its conic classification, standard form, and application in algebra and geometry. Perfect for students and math learners seeking clarity on hyperbolas and quadratic curves.", "---", "Strategically structured with technical clarity and SEO best practices, this article empowers readers to confidently approach similar equations and appreciate the beauty of algebraic geometry."]