4x^2 + 4y^2 + 4z^2 = 16 - 8z + z^2 \Rightarrow 4x^2 + 4y^2 + 3z^2 + 8z - 16 = 0.
["Optimize Your Understanding: Solving the Quadratic Equation 4x² + 4y² + 3z² + 8z – 16 = 0", "---", "Introduction", "Working with polynomial equations is fundamental in algebra and advanced mathematics. Today, we explore a key transformation that simplifies the equation\n4x² + 4y² + 3z² + 8z – 16 = 0\nand reveals its geometric significance. This ellipsoid-shaped surface arises frequently in 3D geometry, physics, and optimization. This article breaks down the algebra, simplifies the equation, interprets its meaning, and discusses applications—helping you master this transformation effortlessly.", "---", "### Step 1: Understanding the Original Equation", "Start with:", "$$\n4x^2 + 4y^2 + 3z^2 + 8z - 16 = 0\n$$", "This is a quadratic equation in three variables. Although not linear, it represents a quadratic surface. The goal is to rewrite it into a standard form to identify its shape and key features.", "---", "### Step 2: Combine and Simplify Terms", "Because all terms involving (x^2), (y^2), and (z^2) appear linearly, group them:", "$$\n4x^2 + 4y^2 + (3z^2 + 8z) - 16 = 0\n$$", "Factor out common coefficients for clarity:", "$$\n4x^2 + 4y^2 + 3\left(z^2 + \frac{8}{3}z\right) - 16 = 0\n$$", "---", "### Step 3: Complete the Square for the z-Term", "Focus on the (z)-terms:\n$$\nz^2 + \frac{8}{3}z\n$$", "To complete the square:\n- Take half of the coefficient of (z): (\frac{8}{3} \div 2 = \frac{4}{3})\n- Square it: (\left(\frac{4}{3}\right)^2 = \frac{16}{9})", "Now rewrite:", "$$\nz^2 + \frac{8}{3}z = \left(z + \frac{4}{3}\right)^2 - \frac{16}{9}\n$$", "Substitute back into the main equation:", "$$\n4x^2 + 4y^2 + 3\left[\left(z + \frac{4}{3}\right)^2 - \frac{16}{9}\right] - 16 = 0\n$$", "Distribute the 3:", "$$\n4x^2 + 4y^2 + 3\left(z + \frac{4}{3}\right)^2 - \frac{48}{9} - 16 = 0\n\Rightarrow\n4x^2 + 4y^2 + 3\left(z + \frac{4}{3}\right)^2 - \frac{16}{3} - 16 = 0\n$$", "Convert 16 to thirds:\n$$\n-16 = -\frac{48}{3}, \quad \ ext{so total constant: } -\frac{16 + 48}{3} = -\frac{64}{3}\n$$", "Thus:", "$$\n4x^2 + 4y^2 + 3\left(z + \frac{4}{3}\right)^2 = \frac{64}{3}\n$$", "---", "### Step 4: Normalize to Standard Form", "Divide both sides by (\frac{64}{3}):", "$$\n\frac{4x^2}{\frac{64}{3}} + \frac{4y^2}{\frac{64}{3}} + \frac{3\left(z + \frac{4}{3}\right)^2}{\frac{64}{3}} = 1\n$$", "Simplify each term:", "- (\frac{4x^2}{\frac{64}{3}} = \frac{12x^2}{64} = \frac{3x^2}{16})\n- (\frac{4y^2}{\frac{64}{3}} = \frac{12y^2}{64} = \frac{3y^2}{16})\n- (\frac{3\left(z + \frac{4}{3}\right)^2}{\frac{64}{3}} = \frac{9\left(z + \frac{4}{3}\right)^2}{64})", "Final standard form:", "$$\n\frac{x^2}{\frac{16}{3}} + \frac{y^2}{\frac{16}{3}} + \frac{\left(z + \frac{4}{3}\right)^2}{\frac{64}{27}} = 1\n$$", "This represents an ellipsoid centered at ((0, 0, -\frac{4}{3})) with semi-axis lengths:", "- Along (x) and (y): (\sqrt{\frac{16}{3}} = \frac{4}{\sqrt{3}})\n- Along (z): (\sqrt{\frac{64}{27}} = \frac{8}{3\sqrt{3}})", "---", "### Step 5: Geometric & Practical Implications", "This ellipsoid describes a symmetric 3D surface centered at (z = -\frac{4}{3}), open along the vertical axis due to the (z)-term adjustment. The factor (\frac{64}{27}) indicates taller vertical stretching in (z), highlighting how constants and completions fundamentally shift planar or curved forms into canonical geometric models.", "Understanding this transformation helps in:", "- Visualizing spatial surfaces in computer graphics\n- Modeling physical phenomena like gravitational potentials\n- Optimizing constraints in multivariable calculus", "---", "### Conclusion", "The original equation\n$$\n4x^2 + 4y^2 + 3z^2 + 8z - 16 = 0\n$$\ntransforms elegantly into the ellipsoid\n$$\n\frac{x^2}{\frac{16}{3}} + \frac{y^2}{\frac{16}{3}} + \frac{\left(z + \frac{4}{3}\right)^2}{\frac{64}{27}} = 1\n$$\nvia completing the square and normalization. This process reveals the surface’s ellipsoidal shape and key descriptive parameters. Mastering this technique empowers deeper insights into nonlinear equations and their geometric interpretations.", "---", "Keywords:\nQuadratic surface, ellipsoid, completing the square, 4x² + 4y² + 3z² + 8z – 16 = 0, algebraic simplification, coordinate geometry, parametric surfaces, mathematical modeling", "---", "Meta Description:\nUnlock a step-by-step guide to simplify and interpret the 4x² + 4y² + 3z² + 8z – 16 = 0 equation into standard ellipsoid form. Learn how completing the square reveals its geometric shape and key properties."]
