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Consider the cubic equation $x^3+d=bx^2$ with $ b,d > 0 $.

The question is to give a geometric solution to this equation by interesting two conic sections.

In class, our teacher showed us how to construct the geometric solution of functions like $x^3+ax=d$, in which the solution is the intersections of a parabola and a circle. However, he didn't show how he got the exact function of the parabola and the circle. So I'm thinking maybe the answer is different since we have a higher order $x^2$? Is there a general way that we can solve this problem? Any hints will be very helpful!

HeroZhang001
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user19170731
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  • The simplest way to do this using modern methods is going to be to write the cubic in depressed form, i.e. $x^3 + ax + b$. Every cubic can be changed into this from, with the roots merely being a constant horizontal shift from the original roots (so just shift the geometric picture by this amount to compensate). By symmetry (eg flipping over axes) we can replace $b$ by $-b$ wlog, so it suffices to solve $x^3 + ax = b$ which is what your teacher did. – Brevan Ellefsen Feb 28 '23 at 00:30
  • "What doth the Mathematician-Astronomer-Poet buy half so precious as he calculates?" See, for instance: https://www.raymaps.com/index.php/omar-khayyams-solution-to-cubic-equations/ , https://math-physics-problems.fandom.com/wiki/Omar_Khayyam%27s_Cubic , https://www.youtube.com/watch?v=RY2DJdMAaJ0 . Khayyam also used a method involving a hyperbola and a circle. –  Feb 28 '23 at 01:01
  • https://www.youtube.com/watch?v=FG5jCgIdttM – Will Jagy Feb 28 '23 at 01:14

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