x^2 + (a - x)^2 + x^2 = 2a^2 \quad \Rightarrow \quad x^2 + a^2 - 2ax + x^2 + x^2 = 2a^2 \quad \Rightarrow \quad 3x^2 - 2ax + a^2 = 2a^2 \quad \Rightarrow \quad 3x^2 - 2ax - a^2 = 0

["# Solving the Quadratic Equation: A Step-by-Step Guide to\nx² + (a − x)² + x² = 2a²", "---", "Understanding the equation\nThe equation x² + (a − x)² + x² = 2a² is a classic algebra problem involving quadratic expressions. At first glance, it may seem complex, but with a clear breakdown, you’ll discover it simplifies neatly into a standard quadratic form:\n3x² − 2ax − a² = 0", "This transformation allows us to apply proven methods for solving quadratics efficiently. Let’s explore each step carefully.", "---", "## Step-by-Step Derivation", "Start with the original equation:\n[\nx^2 + (a - x)^2 + x^2 = 2a^2\n]", "Step 1: Expand (a − x)²\n[\n(a - x)^2 = a^2 - 2ax + x^2\n]", "Substitute into the equation:\n[\nx^2 + (a^2 - 2ax + x^2) + x^2 = 2a^2\n]", "Step 2: Combine like terms\nLeft-hand side becomes:\n[\nx^2 + a^2 - 2ax + x^2 + x^2 = 3x^2 - 2ax + a^2\n]", "So now we have:\n[\n3x^2 - 2ax + a^2 = 2a^2\n]", "Step 3: Move all terms to one side\nSubtract 2a² from both sides:\n[\n3x^2 - 2ax + a^2 - 2a^2 = 0 \Rightarrow 3x^2 - 2ax - a^2 = 0\n]", "---", "## Solving the Quadratic Equation:\n3x² − 2a x − a² = 0", "This is a standard quadratic equation of the form:\n[\nAx^2 + Bx + C = 0\n]\nwith:\n- A = 3\n- B = −2a\n- C = −a²", "### Apply the quadratic formula:\nThe quadratic formula is:\n[\nx = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\n]", "Substitute A, B, and C:\n[\nx = \frac{-(-2a) \pm \sqrt{(-2a)^2 - 4(3)(-a^2)}}{2(3)}\n]", "Simplify:\n[\nx = \frac{2a \pm \sqrt{4a^2 + 12a^2}}{6} = \frac{2a \pm \sqrt{16a^2}}{6}\n]", "[\nx = \frac{2a \pm 4|a|}{6}\n]", "Since √(16a²) = 4|a|, we consider two cases based on the sign of a:", "---", "### Case 1: a > 0\nThen |a| = a:\n[\nx = \frac{2a \pm 4a}{6}\n]", "- First solution:\n[\nx = \frac{2a + 4a}{6} = \frac{6a}{6} = a\n]", "- Second solution:\n[\nx = \frac{2a - 4a}{6} = \frac{-2a}{6} = -\frac{a}{3}\n]", "---", "### Case 2: a < 0\nThen |a| = −a (since a is negative):\n[\nx = \frac{2a \pm 4(-a)}{6} = \frac{2a ∓ 4a}{6}\n]", "- First solution:\n[\nx = \frac{2a - 4a}{6} = \frac{-2a}{6} = -\frac{a}{3}\n]", "- Second solution:\n[\nx = \frac{2a + 4a}{6} = \frac{6a}{6} = a\n]", "In both cases, the solutions remain:\n[\nx = a \quad \ ext{and} \quad x = -\frac{a}{3}\n]", "> Note: Since a is typically taken as a positive constant in such problems, but the formal solutions are valid for both signs of a.", "---", "## Where does this equation apply?", "This expression arises in optimization problems, minimization of quadratic cost functions, and symmetry-based modeling in physics and engineering — for example, balancing two force terms or minimizing energy expressions involving squared deviations.", "---", "## Final Answer", "The solutions to the equation\n[\nx^2 + (a - x)^2 + x^2 = 2a^2\n]\nare:\n[\n\boxed{x = a} \quad \ ext{and} \quad \boxed{x = -\frac{a}{3}}\n]", "---", "## Why This Matters", "Mastering such algebraic transformations not only helps solve equations but also strengthens your problem-solving toolkit for advanced math, physics, and engineering challenges. Understanding how expansions, combining, and applying the quadratic formula lead from complex forms to clean solutions is invaluable.", "---", "Keywords:\nx² + (a − x)² + x² = 2a², quadratic equation solved, algebra steps, solve 3x² − 2ax − a² = 0, quadratic formula application, simplify algebraic equation, x solution derivation, math problem solving, high school algebra, intermediate algebra, quadratic optimization", "---", "Ready to tackle your next equation? Open the steps — algebraic transformation is your guide."]