a geometric interpretation for the quadratic equation

I just discovered one of those gems. It was right there, a bit hidden, a bit shy. But it was so cute! See:

Given a quadratic polynomial , you surely know how to solve the equation from school. Most probably you memorized it:

But you can make a very cool geometrical interpretation out of it – that when solving the quadratic (or cubic) equation what we are really doing is looking for a rectangle (or a box). I had never seen it until last night when I was thinking of and toying with quadratic and cubic equations.

To see it, lets first note that in the case of the quadratic we are looking for two solutions/roots (which we’ll call r and s), meaning we can write our quadratic function as too. Now, if we multiply distributively, group the terms by powers of x and match those to the coefficients of the original equation, we get that

So, one could interpret that we are looking for a rectangle with sides r and s with area c and perimeter -2b. To find what the geometrical interpretation is for the famous quadratic equation solution, we can start by looking at the discriminant . This is clearly telling as to take b, ie, the segments r and s, and form a square with them, then substract four rectangles of size c or rs:


The remaining rectangle in the center is the discriminant (b² – 4c, “one square of side b minus 4 rectangles c“). We are interested in its side , because we have to add half its length to the central point b/2 of the segment r+s to reach r and s for = (r-s)/2.

In the case of the cubic equation similar things happen. When we rewrite f(x) as and we multiply and group terms, we arrive to:

which sort of tells us that the solution to the cubic is actually a box with sides r, s and t that add to -b, that has a surface area of 2c and the same volume as a cube -d.

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