The Persian polymath Omar al-Khayyami developed a geometric technique for finding the positive real roots of cubic and quartic equations. In modern notation, his method for solving cubics can be described as follows:
To solve the cubic equation1. Verify that al-Khayyami's method for solving cubic equations actually does find positive real roots, if such exist. [10]
intersect the hyperbola y = bc/x + b with the circle (x+(a+c)/2)2 = (a-c)2/4 and discard the point (-c,0).
- x3 + ax2 + b2x + b2c = 0
Correction: the equation of the hyperbola should y = bc/x + b instead of y = bc/bx + b, as originally stated.
2. Find al-Khayyami's method for solving quartic equations or devise a geometrical method of your own for doing so, and verify that it works. [10]
Bonus. One of al-Khayyami's friends became a high government official and another became the leader of a (then) violent religious sect. Who were they? [1]
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