On clay tablets dated between 1800 BC and 1600 BC, the ancient Babylonians left the earliest evidence of the discovery of quadratic equations, and also gave early methods for solving them.

Indian mathematician Baudhayana who wrote a Sulba Sutra in ancient India circa 8th century BC first used quadratic equations of the form

$ax^2=c$ and $ax^2+bx=c$

and also gave methods for solving them.

Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used the method of completing the square to solve quadratic equations with positive roots, but did not have a general formula.

Euclid, a Greek mathematician, produced a more abstract geometrical method around 300 BC.

The first mathematician to have found negative solutions with the general algebraic formula was Brahmagupta (India, 7th century).

Muḥammad ibn Mūsā al-Ḵwārizmī (Persia, 9th century) developed a set of formulae that worked for positive solutions.

Bhaskara II (1114-1185), an Indian mathematician-astronomer, solved quadratic equations with more than one unknown and is considered the originator of the equation.

Shridhara (India, 9th century) was one of the first mathematicians to give a general rule for solving a quadratic equation.