Editing Cubic equation (section)

===Omar Khayyám's solution===
[[File:Omar Kayyám - Geometric solution to cubic equation.svg|thumb|right|200px|Omar Khayyám's geometric solution of a cubic equation, for the case {{math|''m'' {{=}} 2}}, {{math|''n'' {{=}} 16}}, giving the root {{math|2}}. The intersection of the vertical line on the {{mvar|x}}-axis at the center of the circle is happenstance of the example illustrated.]] 
For solving the cubic equation {{math|''x''<sup>3</sup> + ''m''<sup>2</sup>''x'' {{=}} ''n''}} where {{math|''n'' > 0}}, [[Omar Khayyám]] constructed the parabola {{math|''y'' {{=}} ''x''<sup>2</sup>/''m''}}, the circle that has as a diameter the [[line segment]] {{math|[0, ''n''/''m''<sup>2</sup>]}} on the positive {{mvar|x}}-axis, and a vertical line through the point where the circle and the parabola intersect above the {{mvar|x}}-axis. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the {{mvar|x}}-axis (see the figure).

A simple modern proof is as follows. Multiplying the equation by {{math|''x''/''m''<sup>2</sup>}} and regrouping the terms gives 
<math display="block">\frac{x^4}{m^2}= x\left(\frac{n}{m^2}-x\right).</math>
The left-hand side is the value of {{math|''y''<sup>2</sup>}} on the parabola. The equation of the circle being {{math|''y''<sup>2</sup> + ''x''(''x'' − {{sfrac|''n''|''m''<sup>2</sup>}}) {{=}} 0}}, the right hand side is the value of {{math|''y''<sup>2</sup>}} on the circle.