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I am well aware of proofs of the quadratic formula that show, by completing the square and other methods, that the quadratic formula is what it is. I have been scouring the Internet and other resources I have and have not made a huge dent into why the quadratic formula works geometrically.

For example, from the geometry of the parabola, why is the axis of symmetry $ \frac{-b}{2a} $? (What does $ b $ mean geometrically? $ c $ represents the vertical shift from the origin and $ a $ refers to the "up or down pointing" of the parabola/maximum or minimum. I think it $ b $ has something to do with the vertex, hence completing the square.) I remember seeing something about this at one point, but I'm not sure where I put the paper I have.

Also, why is the distance from the axis of symmetry to the roots the square root of the discriminant divided by $ 2a $? Is there some geometrical reasoning for this that I cannot see?

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This isn't much more geometric, but consider that if the parabola $y = ax^2$ is moved to the right by $h$ and up by $k$, then the equation becomes $y = a(x-h)^2 + k$, which is just the same as $y = ax^2 - 2ahx + ah^2 + k$, so $h$ can be retrieved from $b$ by dividing by $-2a$.

Also, the roots are where $x - h = \pm\sqrt{-k/a}$, and since $k = c - ah^2 = c - b^2/4a$, it follows that $x - h = \pm\sqrt{b^2-4ac}/2a$.

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  • $\begingroup$ I guess I'm looking to see if there is a reason for dividing by $2a$. What's magic about that number from a visual standpoint? What's magic about $ b^2 - 4ac $ from a visual standpoint? That's the kind of thing I'm attempting to address. It seems that the numbers come from completing the square, but how do these numbers come up from a parabola? $\endgroup$ – ZachTheRiah Feb 3 '15 at 12:26

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