Why does the quadratic formula give solution to quadratic equation

Let the quadratic equation be of the form: ax² + bx + c = 0.

Now by completing the square ( which can be done numeorous ways) get: a(x - b / 2a)² - a(b / 2a)² + c = 0.

Then rearrange equation:

a(x - b / 2a)² = a(b / 2a)² + c, x - (b / 2a) = +- sqrt( (ab²  - 4a²c) / 4a³) and hence x = (- b  +- sqrt(b²  - 4ac)) / 2a