The Babylonian method to solve quadratic equations

You will never forget how to solve quadratic equations once you see how Babylonians did it back in 2000 B.C. It’s striking that these methods date back 4000 years!

Quadratic equations are everywhere. You see some manifestation of it, often as parabolas, every single day. For example:

Bouncing ball in a parabolic trajectory
Satellite dish parabola
Sydney harbor bridge parabola
Water trajectory parabola

Babylonians, who lived in the fertile region between Euphrates River and Tigris River were also interested in quadratic equations. Consider this cute puzzle from 2000 B.C.

I have added the area and two-thirds of [ the side of ] my square and it is 35. What is the side of my square?

Babylon puzzle from 2000 B.C.

In modern notation, this puzzle corresponds to solving for x in

    \[x^2 + \frac{2x}{3} = 35\]

Babylonians had a geometric way of solving such equations. Let me explain. To solve a general quadratic equations of the form

    \[x^2 + px = q\]

Babylonians considered the following square.

Rendered by

The area of the larger square with side length x + \frac{p}{2} is the sum of the areas of the individual squares and rectangles shaded in various colors above. Therefore,

    \begin{eqnarray*}(x + \frac{p}{2}) ^2 & = x^2 + 2 \frac{p}{2} x + \frac{p^2}{4} \\&= x^2 + px + \frac{p^2}{4}\end{eqnarray*}

Now, x^2 + px = q. This gives (x + \frac{p}{2}) ^2 = q + \frac{p^2}{4}

Therefore, x = \sqrt{q + \frac{p^2}{4}} - \frac{p}{2} 😀

Here is a trivia if you enjoyed reading this far.

Why did the Babylonians represent a general quadratic equation in the form x^2 + px = q, as opposed to the contemporary form ax^2 + bx + c = 0?

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2 thoughts on “The Babylonian method to solve quadratic equations”

  1. Sujit,

    I read your post about the Babylonian method for solving quadratic equations and I notice that his post shows the LaTex thread (code), but not the actual formulas. Can you send me the actual post so that I might see it as you intended? I read something about this method and I want to look into it further. I do use Latex, so I know what you are doing here, but find my version rather distracting.

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