# Geometry of the Quadratic Formula

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?

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 $x - h = \pm\sqrt{b^2-4ac}/2a$.
• 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? – ZachTheRiah Feb 3 '15 at 12:26