Solve the equation x^3=3x+4 like Cardano Cardano\'s Method for Solving the Cubic

Question

Solve the equation x^3=3x+4 like Cardano

Cardano's Method for Solving the Cubic Example: Solve an equation of the form x3 cx d. O Cardano asserts that the solution will be of the form x il v, where uv c/3 and u v3 B d By using x3 (u v)3 u3 3u2v 3uv2 v3 Cardano is searching for the right values of u and v to make his solution x. O Having demonstrated his approach geometrically, Cardano gives a verbal version of the formula: He does not describe how he found the formula. (Valuable secrets, you know.) However, Babylonian and Islamic sources do suggest that this method of solving a cubic equation (geometrically, not algebraically) were already known O He also mentions the difficulty of finding a real solution if CH This would have meant the square root of a negative quantity.

Explanation / Answer

x3 = 3x + 4

So c = 3 d = 4

x = u+v, uv = 3/3 = 1 u3+v3 = 4

Sustituting in Cardiano's formula,

x = 3 ( d/2 + ( (d/2)2 - (c/3)2 ) ) + 3 ( d/2 - ( (d/2)2 - (c/3)2 ) )

x = 3 ( 4/2 + ( (4/2)2 - (3/3)2 ) ) + 3 ( 4/2 - ( (4/2)2 - (3/3)2 ) )

x = 3 ( 2 + (22 - 12 ) ) + 3 ( 2 - ( 22 - 12 ) )

x = 3 ( 2 + (4 - 1 ) ) + 3 ( 2 - ( 4 - 1 ) )

x = 3 ( 2 + 3 ) + 3 ( 2 - 3 )

This is a root of the equation.

So u = 2 + 3 and v = 2 - 3

The complex roots will be (2 + 3) w + (2 - 3)w2 and  (2 - 3) w + (2 + 3)w2 where w is a cube root of unity.

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