x^2 +b/a +(b/2a)^2 = -c/a + (b/2a)^2
The (b/2a)^2 goes away, since it's on both sides. That leaves (after fixing your typo)
x^2 + bx/a = -c/a
ax^2 + bx = -c
ax^2 + bx + c = 0
However, it appears you want to solve by completing the square
x^2 + b/a x + (b/2a)^2 = (x + b/2a)^2
So that gives
(x + b/2a)^2 = -c/a + b^2/4a2
(x + b/2a)^2 = (b^2-4ac)/(2a)^2
now take square roots
x + b/2a = ±√(b^2-4ac)/2a
x = (-b±√(b^2-4ac)) / 2a
which is the quadratic formula for solving the first equation