The diagonals of a convex quadrilateral are mutually perpendicular. The sum of the lengths of the diagonals is 10. We want to find the maximum possible area of such a quadrilateral.
Let us denote by $x$ and $y$ the lengths of the two diagonals. Then the area of the quadrilateral is the following function of $x$ and $y$:

If we solve for $y$ in terms of $x$, we can reexpress this area as the following function of $x$ alone:

Thus we find that the maximum area is