Math 409/409G History of Mathematics The Babylonian Treatment of Quadratic Equations

The Quadratic Formula The Babylonians had knowledge of the quadratic formula, but not in the form that we know and use it.

In fact, the Babylonians hadn’t yet discovered the concept of a formula. Instead, they used numerical recipes that were equivalent to using a formula. Babylonian Example: To solve “You take 1, the coefficient [of x]. Two thirds of 1 is 0;40. Half of this, 0;20, you multiply by 0;20 and …”

This Babylonian “recipe” would today be stated as: The solution to the quadratic equation is The Babylonians never considered the solution with the negative square root.

When dealing with quadratic equations, the Babylonians always wrote the equation in a form where all numbers were positive and the leading coefficient was 1.

We know the Babylonians had “recipes” for two forms of the quadratic equation. Although we don’t know if they had “recipes” for the other forms, we do know that they had techniques for solving these forms.

Babylonian method of solving Set and Inspiration: The Babylonians were very interested in finding the sides x and y of a rectangle having semi-perimeter a and area b. Such a problem is equivalent to the above settings. The solution to this system is a quadratic equation.

Solving . Set and Set and Note: This setting implies

Solution to So

Babylonian solution to . Set: Then: So

The Babylonian Treatment of Quadratic Equations This ends the lesson on The Babylonian Treatment of Quadratic Equations