Completing the square works because you can change one
parabola into another using standard transformations.
You can scale the parabola.
And you can move the parabola or
translate it. Horizontally.
All vertically.
You can also reflect the parabola. That's like a
scaling, but with the scale factor of minus one.
Now the standard quadratic expression Y equals X squared
gives this parabola.
And any other quadratic will always give a parabola, but it
will be a different size and be in a different place.
So to get this new parabola, we take the standard parabola
and 1st change its eyes instead of Y equals X squared. It's
now Y equals 1/2 X squared.
Next we move the parabola sideways instead of Y equals 1/2
X squared. The ex has become X minus 2.
And finally we move the parabola
vertically. We add on one to the formula and that
gives us the quadratic in completed square form.