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*.