Let $A \in \mathbb{R}^{N \times N}$ and $x \in \mathbb{R}^{N \times 1}$ each be a D4571: Real matrix such that

(i)	$N \in 1, 2, 3, \ldots$ is a D5094: Positive integer

Result R3749: Partition of real square matrix into sum of symmetric and antisymmetric parts allows us to write \begin{equation} A = \frac{A + A^T}{2} + \frac{A - A^T}{2} \end{equation} Multiplying from the left by $x^T$ and from the right by $x$ and applying R3745: Real square matrix antisymmetric part is zero definite, we then have \begin{equation} x^T A x = x^T \left( \frac{A + A^T}{2} \right) x + x^T \left( \frac{A - A^T}{2} \right) x = x^T \left( \frac{A + A^T}{2} \right) x \end{equation} $\square$