Math 409/409GHistory 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.

22 4

02

b b acax bx c x

a

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 …”

2 3523 60x x

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.

2x ax b 2

.2 2

a ax b

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.

2 2

2 2

0x ax b x ax b

x b ax x ax b

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.

22

22

2 2

2 2

a ax ax b x b

a ax ax b x b

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.

2x b ax x y a .xy b

2, ( )y a x xy b x a x b x b ax

Solving .

Set and

Set and

Note: This setting implies

2x b ax

x y a .xy b

2

ax z .

2

ay z

.x y a

Solution to

So

Settings

2

2

ax

ay

xy

z

z

b

2 .x b ax

2

2

2

2

2

2

2b

a

azxy b

z

a

b

z

az b

2

2 22

a aaz bx

Babylonian solution to .

Set:

Then:

So

2 2x x 1

2211, ,2 , y zxx zxyy

2

2

3

14

2

12

1 ( )( 2

2

2 )xy z

z

z

z

32

12

12 2x z

This ends the lesson on

The Babylonian Treatment of Quadratic Equations