Cubic equation

In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. An example is the equation

Cardano's method

The solutions can be found with the following method due to Scipione dal Ferro and Tartaglia, published by Gerolamo Cardano in 1545.

Related Topics:
Scipione dal Ferro - Tartaglia - Gerolamo Cardano - 1545

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We first divide the given equation by α3 to arrive at an equation of the form

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:x^3 + ax^2 + bx +c = 0. qquad (1)

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The substitution x = t - a/3 eliminates the quadratic term; in fact, we get the equation

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: t^3 + pt + q = 0, quadmbox{where } p = b - rac{a^2}3 quadmbox{and}quad q = c + rac{2a^3-9ab}{27}. qquad (2)

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This is called the depressed cubic.

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Suppose that we can find numbers u and v such that

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: u^3-v^3 = q quadmbox{and}quad uv = rac{p}{3}. qquad (3)

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A solution to our equation is then given by

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:t = v - u,

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as can be checked by directly substituting this value for t in (2), as a consequence of the third order binomial identity

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: (v-u)^3+3uv(v-u)+(u^3-v^3)=0 .

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The system (3) can be solved by solving the second equation for v, which gives

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: v = rac{p}{3u}.

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Substituting this in the first equation in (3) yields

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: u^3 - rac{p^3}{27u^3} = q.

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This can be seen as a quadratic equation for u3. If we solve this equation, we find that

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: u=sqrt{{qover 2}pm sqrt{{q^{2}over 4}+{p^{3}over 27}}}. qquad (4)

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Since t = v − u and t = x + a/3, we find

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:x=rac{p}{3u}-u-{aover 3}.

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Note that there are six possibilities in computing u with (4), since there are two solutions to the square root (pm), and three complex solutions to the cubic root (the principal root and the principal root multiplied by -1/2 pm isqrt{3}/2). However, which sign of the square root is chosen does not affect the final resulting x, although care must be taken in two special cases to avoid divisions by zero. First, if p = 0, then one should choose the sign of the square root that gives a nonzero value for u, i.e. u = sqrt{q}. Second, if p = q = 0, then we have the triple real root x = −a/3.

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