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.