Part One: History of the Quadratic
Even though the Babylonians didn't have any notion of what an 'equation' is, they found the first algorithmic approaches to problems, which would give rise to a quadratic equation today. Their method is essentially one of completing the square. However all Babylonian problems had answers which were positive (more accurately unsigned) quantities since the usual answer was a length.
The first known solution of a quadratic equation is the one given in the Berlin papyrus from the Middle Kingdom (ca. 2160-1700 BC) in Egypt.
This problem reduces to solving
x2 + y2 = 100
y = 3/4 x
In about 300 BC Euclid developed a geometrical approach which, although later mathematicians used it to solve quadratic equations, amounted to finding a length which in our notation was the root of a quadratic equation. Euclid had no notion of equation, coefficients etc. but worked with purely geometrical quantities. His Data contains three problems involving quadratics.
In his work Arithmetica, the Greek mathematician Diophantus (ca. 210-290) solved the quadratic equation, but giving only one root, even when both roots were positive.
Hindu mathematicians took the Babylonian methods further. Aryabhata (475 or 476-550) gave a rule for the sum of a geometric series that shows knowledge of the quadratic equations with both solutions.