f(x+y) + f(x-y) = c(x+y)^2 + c(x-y)^2 = c(x^2 + 2xy + y^2 + x^2 - 2xy + y^2) = c(2x^2 + 2y^2) = 2cx^2 + 2cy^2 = 2f(x) + 2f(y)

["# Exploring the Functional Equation: Proving f(x+y) + f(x−y) = c(x+y)² + c(y−x)² and Its Mathematical Significance", "Functional equations play a pivotal role in advanced mathematics, connecting abstract function behavior with algebraic identities. One particularly instructive equation—f(x+y) + f(x−y) = c(x+y)² + c(y−x)²—offers insight into evenness, symmetry, and quadratic functions. In this article, we explore this elegant identity, break down its derivation, and examine its implications in mathematical analysis.", "---", "## The Functional Equation: An Algebraic Identity", "Consider the equation:", "[\nf(x+y) + f(x-y) = c(x+y)^2 + c(y-x)^2\n]", "At first glance, the right-hand side appears as a sum of two expanded quadratic expressions. Let’s simplify the right-hand side algebraically:", "[\nc(x+y)^2 + c(y-x)^2 = c\left[(x+y)^2 + (y-x)^2\right]\n]", "Expand both squares:", "[\n(x+y)^2 = x^2 + 2xy + y^2\n]\n[\n(y-x)^2 = y^2 - 2xy + x^2\n]", "Adding them:", "[\n(x+y)^2 + (y-x)^2 = (x^2 + 2xy + y^2) + (x^2 - 2xy + y^2) = 2x^2 + 2y^2\n]", "Therefore:", "[\nf(x+y) + f(x−y) = c(2x^2 + 2y^2) = 2c x^2 + 2c y^2\n]", "Now, observe the right-hand side simplifies to 2c times a symmetric expression in x and y. Importantly, the structure strongly suggests that f may encode quadratic behavior.", "---", "## A Closer Look: Hypothesis on f(x)", "Suppose that ( f(x) ) is a quadratic function:\n[\nf(x) = a x^2 + b x + d\n]", "Then:\n[\nf(x+y) = a(x+y)^2 + b(x+y) + d = a(x^2 + 2xy + y^2) + b(x + y) + d\n]\n[\nf(x−y) = a(x−y)^2 + b(x−y) + d = a(x^2 - 2xy + y^2) + b(x - y) + d\n]", "Add them:", "[\nf(x+y) + f(x−y) = a[2x^2 + 2y^2] + b[2x] + 2d = 2a x^2 + 2a y^2 + 2b x + 2d\n]", "Compare with the required right-hand side:", "[\n2c x^2 + 2c y^2 = 2c(x^2 + y^2)\n]", "For identity to hold for all x, y, coefficients must match. Thus:", "- ( 2a = 2c \Rightarrow a = c )\n- ( 2d = 0 \Rightarrow d = 0 ) (constant term must vanish)\n- The linear term ( 2b x ) must be zero for all x → ( b = 0 )", "Hence, the most general quadratic function satisfying the equation is:", "[\nf(x) = c x^2\n]", "---", "## Verifying the Solution", "Let ( f(x) = c x^2 ). Compute:", "[\nf(x+y) + f(x−y) = c(x+y)^2 + c(x−y)^2 = c[(x^2 + 2xy + y^2) + (x^2 - 2xy + y^2)] = c(2x^2 + 2y^2)\n]", "This matches the expanded form ( 2c x^2 + 2c y^2 ), confirming the identity holds.", "---", "## Mathematical Implications and Applications", "### 1. Symmetry and Evenness\nThe result ( f(x+y) + f(x−y) = 2c(x^2 + y^2) ) implies that ( f(x) ) must be an even function when restricted to even shifts. Since this holds for all x, y, the even symmetry of ( f ) is reinforced.", "### 2. Quadratic Form and Linearity of the Functional\nThe expansion eliminates all linear terms and constant offsets, confirming that ( f ) behaves quadratically. This type of functional equation often arises in physics (e.g., energy expressions), geometry (distances squared), and optimization problems involving quadratic forms.", "### 3. Utility in Deriving General Solutions\nThis identity exemplifies a common strategy: assuming a functional form (here quadratic), substituting, and solving for coefficients. Functional equations of this type are frequently used in mathematical olympiads and functional analysis to deduce functional identities cleanly.", "---", "## Conclusion", "The identity ( f(x+y) + f(x−y) = c(x+y)^2 + c(y−x)^2 ) simplifies to ( f(x+y) + f(x−y) = 2c(x^2 + y^2) ), leading naturally to the conclusion that the only quadratic function satisfying the equation is ( f(x) = c x^2 ). This illustrates how algebraic manipulation and symmetry considerations can unlock deep results in functional relations. Known broadly in harmonic analysis, number theory, and differential equations, such identities highlight the elegant interplay between algebra and function behavior.", "Understanding these relationships not only solves specific problems but also strengthens one’s toolkit for tackling complex functional equations in advanced mathematics.", "---", "## Further Reading", "- Functional Equations in Mathematical Olympiads\n- Quadratic Forms and Their Properties\n- Symmetry in Functional Equations: An Introduction\n- Applications of Functional Analysis in Modern Mathematics", "---", "#### Keywords:\nfunctional equation, f(x+y) + f(x−y), quadratic functions, symmetry, c(x+y)², c(y−x)², mathematical identity, algebraic simplification, mathematics education, functional analysis."]