The Oldest Known Equation: How Babylonians Solved Quadratics 4,000 Years Ago

The Babylonian Quadratic Equation That Started It All

Think about it — we have been solving quadratic equations for at least 4,000 years.

The oldest known equation to survive to this day is inscribed on a Babylonian clay tablet approximately 3,800 years old.

The Problem on the Tablet

The tablet poses the following problem:

The length of a rectangle exceeds its width by 7, and the area equals 60. Find the length and the width.

Translated into modern mathematical language, this gives us a system of equations:

x · y = 60
y − x = 7

From which we derive a standard quadratic equation in x:

x² + 7x − 60 = 0

How Babylonians Solved Quadratic Equations Without Formulas

The Babylonians did not use algebraic formulas — instead, they followed a step-by-step algorithm strikingly similar to what we now call “completing the square.” Here is how it worked:

• Take the difference of the sides: 7
• Divide it in half
• Square the result
• Add it to the area: 60
• Take the square root
• Subtract half the difference — this gives the smaller side
• Add 7 — this gives the larger side

Negative numbers had not yet been invented, but for practical geometric problems, positive numbers were more than enough.

The Answer — and a Challenge

Fortunately, no complex numbers were required either.

Answer: length = 12, width = 5.

How did you solve it?

• Worked it out mentally
• Used Vieta’s formulas
• Used the discriminant
• Used a different method entirely