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Rectangle on a cubic (Posted on 2019-03-07) Difficulty: 3 of 5
If a cubic equation has real zeroes +p, -p and 0, and its two horizontal tangent points and two non-zero x-intercepts can be joined together to form a rectangle, then find the ratio of the rectangle’s larger side to its shorter side.

No Solution Yet Submitted by Danish Ahmed Khan    
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If the polynomial is f(x)=x(x-p)(x+p) then the horizontal tangent points are where the derivative equals zero.  That is


x=+/- p/sqrt(3)

The point in the 4th quadrant is (p/sqrt(3), -2p^3/3sqrt(3))

The quadrilateral described is a parallelogram because of its rotational symmetry.  One way to make it a rectangle is to make the four corners equidistant from the origin:

Out of time here... 

The solution is an interesting fourth root:


I didn't finish finding the ratio.  

  Posted by Jer on 2019-03-07 15:57:25
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