You asked "And how was the quadratic
formular proved if not by completing the square?"
One method is in Ray's Algebra (1857)
When an equation is brought to the form ax2 +bx = c it may be reduced
to an equation of the first degree, without dividing by the coefficent
of x2; thus avoiding fractions. If we multiply every term of the
equation ax2+bx = c by four times the coefficent of the first term, and
add to both sides the square of the coefficent of the second term, we
shall have,
4a2x2 + 4abx + b2 = 4ac + b2 Now the first member is a
perfect square, and by extracting the square root of both sides we have
2ax + b = +/-sqrt(4ac+b2) which is an equation of the first degree.
This is called the Hindoo method of solving quadratic equations.
Peace,
Don Cook