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Problem

Figure 1. Omar Khayyam (1048 – 1131 AD)

Omar Khayyam (1048 – 1131 AD) solved cubic equations of the form by transforming the problem into finding the intersection of a circle and a parabola:

where are positive numbers.


Part 1: Determine  in terms of .

Part 2: Use Omar Khayyam’s method to solve the cubic equation  by sketching the two conic sections and locating the intersection points.

Solution

Solution for Part 1

Write the first equation as and the second equation as . Substitute the second equation into the first equation to eliminate , and obtain .

Consequently,

or

Matching terms with the cubic equation gives

Since are positive numbers,

Solution for Part 2

For the equation , and . Thus,

and we need to graph

Figure 2. Parabola and the circle associate d with the cubic equation


Intersection points: ; . When the values of is substituted into , we find one solution to this cubic .

Figure 3. The original cubic equation


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