\Rightarrow (x^2 + y^2 + z^2) - 2x - 2y + 2 = 2

["Title: Solve the Equation: (x² + y² + z²) - 2x - 2y + 2 = 2 — Step-by-Step Explanation", "Mathematics often presents us with equations that may appear complex at first glance, but with proper manipulation, they reveal elegant solutions. One such equation is:", "[\n(x^2 + y^2 + z^2) - 2x - 2y + 2 = 2\n]", "This article will guide you step-by-step through solving this equation, turning it into a simplified form that reveals solutions for variables (x), (y), and (z). Whether you're a student, teacher, or math enthusiast, understanding this process deepens your algebraic skills and enhances problem-solving confidence.", "---", "### Step 1: Simplify the Equation", "Start by simplifying the original equation:", "[\n(x^2 + y^2 + z^2) - 2x - 2y + 2 = 2\n]", "Subtract 2 from both sides to eliminate the constant:", "[\nx^2 + y^2 + z^2 - 2x - 2y = 0\n]", "---", "### Step 2: Rearrange Terms", "Group (x), (y), and (z) terms together for clearer analysis:", "[\nx^2 - 2x + y^2 - 2y + z^2 = 0\n]", "Notice that the (z^2) term stands alone, indicating that (z^2) must compensate for any imbalance.", "---", "### Step 3: Complete the Square for (x) and (y)", "To simplify quadratic expressions, complete the square.", "For the (x)-terms:\n[\nx^2 - 2x = (x - 1)^2 - 1\n]", "For the (y)-terms:\n[\ny^2 - 2y = (y - 1)^2 - 1\n]", "Substitute these back into the equation:", "[\n(x - 1)^2 - 1 + (y - 1)^2 - 1 + z^2 = 0\n]", "Combine constants:", "[\n(x - 1)^2 + (y - 1)^2 + z^2 - 2 = 0\n]", "---", "### Step 4: Final Simplified Form", "Move the constant to the right-hand side:", "[\n(x - 1)^2 + (y - 1)^2 + z^2 = 2\n]", "This equation describes a 3D sphere centered at the point ((1, 1, 0)) with radius (\sqrt{2}). Every point ((x, y, z)) lying on this surface satisfies the original equation.", "---", "### Step 5: General Solution Insight", "Although infinitely many real solutions exist (since this is a surface in 3D space), to find specific solutions, we may fix two variables and solve for the third. For example:", "- If (x = 1) and (y = 1), then:\n [\n z^2 = 2 \Rightarrow z = \pm\sqrt{2}\n ]\n- If (z = 0), then\n [\n (x - 1)^2 + (y - 1)^2 = 2\n ]\n This is a circle in the (xy)-plane centered at ((1,1)) with radius (\sqrt{2}).\n- If (x = y), set (x = y = a), then solve:\n [\n 2(a - 1)^2 + z^2 = 2 \Rightarrow (a - 1)^2 + \frac{z^2}{2} = 1\n ]", "These examples illustrate how structured manipulation enables insight and solution generation.", "---", "### Why This Equation Matters", "Understanding equations like the one above is crucial in geometry, physics, and optimization. Surfaces defined by quadratic expressions model orbits, energy states, and many natural phenomena. Mastering algebraic manipulation empowers you to explore these applications confidently.", "---", "### Summary", "The equation\n[\n(x^2 + y^2 + z^2) - 2x - 2y + 2 = 2\n]\nsimplifies through completing the square to the spherical form:\n[\n(x - 1)^2 + (y - 1)^2 + z^2 = 2\n]\nrevealing a sphere centered at ((1, 1, 0)) with radius (\sqrt{2}). This geometric interpretation opens doors to deeper mathematical exploration.", "---", "Keywords: Solve equation, algebra steps, complete the square, 3D geometry, sphere equation, coordinate geometry, mathematical solution, equation manipulation, geometry insight", "Meta Description:\nExplore how to solve ((x^2 + y^2 + z^2) - 2x - 2y + 2 = 2) step-by-step. Discover the geometric meaning and general solution insights for this 3D sphere equation. Ideal for students and math learners.", "---", "Read more:\n- How to Solve Quadratic Equations by Completing the Square\n- Geometry of Spheres in 3D Space\n- Applications of Algebraic Equations in Science and Engineering"]