Other terms match. Thus, $ h(x) = rac{1}{2}x^2 + bx $. Verify that $ h(x + y) = rac{1}{2}(x + y)^2 + b(x + y) = rac{1}{2}x^2 + xy + rac{1}{2}y^2 + bx + by $, which equals $ h(x) + h(y) + xy $. The general solution is $ h(x) = rac{1}{2}x^2 + bx $, where $ b $ is a constant. The function is $ oxed{h(x) = rac{1}{2}x^2 + bx} $.1.

["# Understanding the Quadratic Form: $ h(x) = \frac{1}{2}x^2 + bx $ and the Identity $ h(x + y) = h(x) + h(y) + xy $", "In mathematics, functions of quadratic form play a vital role in calculus, algebra, and applied modeling. One particularly interesting function is $ h(x) = \frac{1}{2}x^2 + bx $, where $ b $ is a real constant. This article explores the mathematical properties of this function, verifies a key identity involving its additive structure, and explains why it represents the general solution to a class of quadratic functions.", "## The Mathematic Form: $ h(x) = \frac{1}{2}x^2 + bx $", "The function $ h(x) = \frac{1}{2}x^2 + bx $ is a quadratic function with two essential components:\n- A parabolic shape defined by $ \frac{1}{2}x^2 $, which controls the curvature and direction of the graph.\n- A linear drift $ bx $, which shifts the entire parabola vertically depending on the value of $ b $.", "This form is widely recognized in optimization, physics (e.g., projectile motion), and economics for modeling parabolic relationships.", "## Verifying the Identity: $ h(x + y) = h(x) + h(y) + xy $", "To confirm the elegant relationship, we begin with the left-hand side using the given definition:", "$$\nh(x + y) = \frac{1}{2}(x + y)^2 + b(x + y)\n$$", "Expanding the square:", "$$\n(x + y)^2 = x^2 + 2xy + y^2 \Rightarrow \frac{1}{2}(x + y)^2 = \frac{1}{2}x^2 + xy + \frac{1}{2}y^2\n$$", "Adding the linear term:", "$$\nh(x + y) = \frac{1}{2}x^2 + xy + \frac{1}{2}y^2 + bx + by\n$$", "Now compute the right-hand side: $ h(x) + h(y) + xy $", "$$\nh(x) = \frac{1}{2}x^2 + bx,\quad h(y) = \frac{1}{2}y^2 + by\n$$", "So:", "$$\nh(x) + h(y) + xy = \left( \frac{1}{2}x^2 + bx \right) + \left( \frac{1}{2}y^2 + by \right) + xy = \frac{1}{2}x^2 + xy + \frac{1}{2}y^2 + bx + by\n$$", "Both sides match exactly:", "$$\nh(x + y) = \frac{1}{2}x^2 + xy + \frac{1}{2}y^2 + bx + by = h(x) + h(y) + xy\n$$", "This identity demonstrates that $ h(x) = \frac{1}{2}x^2 + bx $ is additive up to a cross-term, capturing both regular quadratic growth and linear interaction between inputs.", "## The General Solution: $ h(x) = \frac{1}{2}x^2 + bx $", "We now show why this is the most general form (up to linear transformation) for functions satisfying:", "$$\nh(x + y) = h(x) + h(y) + xy \quad \ ext{for all real } x, y\n$$", "This functional equation is a variant of Jensen’s equation with a correction term. The extra $ xy $ term reveals a structured dependency on $ x $ and $ y $ beyond standard additive functions.", "To solve it, assume a quadratic polynomial form:\nLet $ h(x) = Ax^2 + Bx $. Plug into the functional equation:", "Left-hand side:\n$$\nh(x + y) = A(x + y)^2 + B(x + y) = A(x^2 + 2xy + y^2) + Bx + By = Ax^2 + 2Axy + Ay^2 + Bx + By\n$$", "Right-hand side:\n$$\nh(x) + h(y) + xy = (Ax^2 + Bx) + (Ay^2 + By) + xy = Ax^2 + Ay^2 + Bx + By + xy\n$$", "Equate both sides:", "$$\nAx^2 + 2Axy + Ay^2 + Bx + By = Ax^2 + Ay^2 + Bx + By + xy\n$$", "Cancel common terms:", "$$\n2Axy = xy\n\Rightarrow 2A = 1 \Rightarrow A = \frac{1}{2}\n$$", "Thus, $ h(x) = \frac{1}{2}x^2 + Bx $. Since $ B = b $ is arbitrary, the general solution is:", "$$\n\boxed{h(x) = \frac{1}{2}x^2 + bx}\n$$", "where $ b $ is a constant determined by initial conditions.", "## Practical Implications", "This form is invaluable in scenarios involving displacement, energy, or error minimization where quadratic growth is present but pairwise interactions introduce a cross-variable term:", "- Physics: Modeling kinetic energy with velocity-dependent forces.\n- Economics: Representing cost or utility functions with additive base costs plus interaction terms.\n- Statistics: Deriving least squares estimators involving quadratic loss.", "## Conclusion: The Unique Structure of $ h(x) $", "The function $ h(x) = \frac{1}{2}x^2 + bx $ is not just any quadratic—it is uniquely characterized by the identity\n$$\nh(x + y) = h(x) + h(y) + xy\n$$\nembodying both pure quadratic behavior and a symmetric pairwise interaction. This makes it the general solution to a class of functions central in applied mathematics.", "Boxed final answer:\n$$\n\boxed{h(x) = \frac{1}{2}x^2 + bx}\n$$"]