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Solving Roots of Cubic Equations Using Paper

Nov 17, 2007

Yes, you read that right. I was amazed when my Math 210.2 professor demonstrated how to do it. I was itching to find the proof on why this technique works. With this, you can find the solutions of this equation:

x^3 + bx^2 + cx + d = 0

Here are the steps:

  1. You need a sheet of tracing paper or onion skin paper. Make a Cartesian plane on this sheet.
  2. Locate the points P(0,1) and Q(-b+d,c). Also, draw the lines y=-1 and the line x=-b-d.
  3. Fold the sheet so that P lies on the line y=-1 and Q lies on the line x=-b-d. You should be able to find at least one such fold line. In some cases, it is possible to find three ways of doing this. (This is the reason why you need tracing paper or onion skin paper, since it will be easier to see if the points will indeed lie on the lines indicated.)
  4. The x-intercept of these fold lines will tell you the roots of the cubic equation given above.

It really works, I’ve tried it twice myself. Have fun, and amaze your friends in the process. 🙂

[Thanks to Texify for the \LaTeX equations inserted in this entry.]

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