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Completing the square works
because you can change one

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parabola into another using
standard transformations.

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You can scale the parabola.

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And you can move the parabola or

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translate it. Horizontally.

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All vertically.

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You can also reflect the
parabola. That's like a

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scaling, but with the scale
factor of minus one.

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Now the standard quadratic
expression Y equals X squared

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gives this parabola.

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And any other quadratic will
always give a parabola, but it

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will be a different size and be
in a different place.

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So to get this new parabola,
we take the standard parabola

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and 1st change its eyes instead
of Y equals X squared. It's

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now Y equals 1/2 X squared.

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Next we move the parabola
sideways instead of Y equals 1/2

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X squared. The ex has become
X minus 2.

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And finally we move the parabola

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vertically. We add on one
to the formula and that

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gives us the quadratic in
completed square form.