I read that in ancient times the quadratic equation of this kind $$x^2+10x=39$$ had been solved long ago. I read that this kind of equation originated in the geometric question of "Given an area of 39, find the side x s.t. the square of x plus the rectangle of x and 10 fit the area."

But this condition "Given...area" seems artificial to me.

My question is: How was this condition related to the actual needs of the real world in ancient times?

Is there any evidence that this kind of condition REALLY grows out of some geometric construction problem for the base of some temple or something like that?

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    $\begingroup$ It is likely that people pursued quadratic problems for the sake of intellectual challenge rather practical need. Here is a discussion of this for the case of the Pythagorean Theorem. $\endgroup$ Sep 22, 2015 at 16:28

1 Answer 1


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.

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    $\begingroup$ Conifold: Thank you very much for your detailed answer! That's exactly what I want to know! Authors of math books in modern times may benefit from ancient Babylonians for their natural demands of the quadratic equations rather than the artificial high-tech examples. Sorry for the delay, I am on the other half of the planet :-) $\endgroup$
    – user2927
    Sep 23, 2015 at 13:40

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