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( (ab- 4a2c) / 4a3) and hence x = (- b  +- sqrt(b- 4ac)) / 2a

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