The Sum of Its Parts

One says that 10 is divided into three parts and if the small part is multiplied by itself and added to the middle one multiplied by itself the result is the large one multiplied by itself, and when the small part is multiplied by the large part, it equals the middle part multiplied by itself. Find the parts.

Kitab fi al-jabr wa al-muqabala (Book of Algebra), Abu Kamil, c. 900

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The answer is: \(5-\displaystyle\sqrt{\sqrt{3125}-50}\approx 2.57066, \left(\sqrt{\dfrac{1}{2}+\dfrac{\sqrt{5}}{2}}\right)\cdot \left(5-\sqrt{\sqrt{3125}-50}\right) \approx 3.26993\), and \(\left(\dfrac{1}{2}+\dfrac{\sqrt{5}}{2}\right)\cdot \left(5-\displaystyle\sqrt{\sqrt{3125}-50}\right) \approx 4.15941\)

"The Sum of Its Parts," Convergence (July 2006)

Convergence