For a broader perspective see How was geometry historically used to solve polynomial equations? For early practical problems that would lead (today) to quadratic equations see e.g. Friberg's discussion of cuneiform tablet YBC 3879 (c. 2000 BC), a judicial document from third Sumerian Ur period, that describes field division problems leading to quadratic equations.
"The field plan shows a nearly square field, apparently a “temen”... A temen is a regularly shaped field, typically made up of one or several rectangles or trapezoids... In connection with a field plan, the word appears to refer to the initial form of a field, before various modifications. In the field plan on the reverse of YBC 3879, six fields of various shapes (three triangles and three (quasi-) trapezoids) have been removed from the temen, leaving as residue an irregularly shaped field".
The geometry in YBC 3879 is rather messy, "the area B that was computed above is also equal to the area of the rectangle with the sides a and u1' minus the area of the right triangle with the sides a and b = f · a", and the OP example seems like a simplified problem of this sort. Babylonians, Egyptians, etc., also had to deal with dividing fields into simple shapes.
Another early occurence is in the Berlin papyrus 6619 (c.1800 BC) with the problem stated as
"You are told the area of a square of 100 square cubits is equal to that of two smaller squares, the side of one square is 1/2 + 1/4 of the other. What are the sides of the two unknown squares?"
The equation is solved by false position, a common maneuver in the absence of notation for unknowns. It could also be motivated by field division, but not necessarily. It is known that papyri were written by Egyptian scribes and contained practice problems for training other scribes in their craft. Some problems were explicitly related to construction, such as reckoning a pyramid in Rhind papyrus, some not, and were specifically meant to teach them how to perform certain computations. Many practical problems involved finding Aha (unknowns) by false position, so mastering this technique was essential.
