** Problem **

**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

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