The found ancient Babylonian clay tablets aged about between 1800 and 1600 years before the Christian Era are the earliest evidence about quadratic equation research. Ways of some solving of quadratic equation are stated on these tablets.
An ancient Indian mathematician Baudhayana in VIII century before the Christian Era used quadratic equations in the form of
ax2 = c and ax2 + bx = cand gave their solution methods for the first time.
Babylonian mathematicians since IV century before the Christian Era and Chinese mathematicians approximately since II century before the Christian Era used method of square complement for solution of equation with positive roots. Euclid devised more general geometrical solution method about 300 years before the Christian Era.
The first mathematician who discovered solution of equation with negative roots in the form of algebraic formula was Brahmagupta (India, VII century before the Christian Era).
You can use Vieta theorem for roots finding of quadratic equation.