I just did this by hand with no algebra and no calculator, just pen and paper, doing all the arithmetic in my head, plotting just seven points and estimating the answer visually.  It took less than one minute.  Conclusion: For 0.83(x − 6.3)2 + 4.4, the imaginary part of the root is a little bit more than 2.  That is all trivial.  Algebra is used to explain why this works, or, if you like, to prove that it works.  No algebra is needed in order to do what I just did.  And so what if you couldn't do all that?  The article still explains the geometric relationship between the parabola and the locations of the roots.  That is the point.