Completing the square won't magically make a quadratic factorise over the reals. You WILL however be able to factorise over the complex numbers, which will still give complex roots.if I complete the square on the same polynomial which makes it
factorable does this process give it real roots?
Any quadratic, whether its coefficients are real or not, can be factored- but NOT necessarily with integer or even real coefficients.
Take as a generic example. First, factor out the "a". . Now "complete the square"- add and subtract : . The point of that is that , a perfect square. And of course, .
That is, we can write
And now we use the fact that to write
That is factored into a number times two linear factors. With those square roots, those numbers may not be integers or even real numbers.