quartic formula
QuarticFormula
2002-01-22 00:21:25
2005-07-07 23:28:48
Theorem
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The four roots $r_1, r_2, r_3, r_4$ of a quartic polynomial equation
$x^4 + ax^3 + bx^2 + cx + d = 0$ are given by
\begin{eqnarray*}
r_1 & = &
{\frac{-a}{4} - \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} +
\frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) }
{3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd +
{\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} \right) }^{\frac{1}{3}}} +
\left(\frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd + {\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} }}
{54}\right)^\frac{1}{3}}} -
\frac{1}{2}{\sqrt{\frac{a^2}{2} - \frac{4b}{3} -
\frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) }
{3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd +
{\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} \right) }^{\frac{1}{3}}} -
\left(\frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd + {\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} }}
{54}\right)^\frac{1}{3} -
\frac{-a^3 + 4ab - 8c}
{4{\sqrt{\frac{a^2}{4} - \frac{2b}{3} +
\frac{2^{\frac{1}{3}}
\left( b^2 - 3ac + 12d \right) }{3
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd +
{\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 +
27a^2d - 72bd \right) }^2}} \right) }^
{\frac{1}{3}}} +
\left( \frac{{ 2b^3 - 9abc + 27c^2 +
27a^2d - 72bd +
{\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 +
27a^2d - 72bd \right) }^2}} }
}{54}\right)^\frac{1}{3}}}}}}} \\
r_2 & = &
{\frac{-a}{4} - \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} +
\frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) }
{3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd +
{\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} \right) }^{\frac{1}{3}}} +
\left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd + {\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} }}
{54}\right)^\frac{1}{3}}} +
\frac{1}{2}{\sqrt{\frac{a^2}{2} - \frac{4b}{3} -
\frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) }
{3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd +
{\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} \right) }^{\frac{1}{3}}} -
\left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd + {\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} }}
{54}\right)^\frac{1}{3} -
\frac{-a^3 + 4ab - 8c}
{4{\sqrt{\frac{a^2}{4} - \frac{2b}{3} +
\frac{2^{\frac{1}{3}}
\left( b^2 - 3ac + 12d \right) }{3
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd +
{\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 +
27a^2d - 72bd \right) }^2}} \right) }^
{\frac{1}{3}}} +
\left( \frac{{ 2b^3 - 9abc + 27c^2 +
27a^2d - 72bd +
{\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 +
27a^2d - 72bd \right) }^2}} }
}{54}\right)^\frac{1}{3}}}}}}} \\
r_3 & = &
{\frac{-a}{4} + \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} +
\frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) }
{3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd +
{\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} \right) }^{\frac{1}{3}}} +
\left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd + {\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} }}
{54}\right)^\frac{1}{3}}} -
\frac{1}{2}{\sqrt{\frac{a^2}{2} - \frac{4b}{3} -
\frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) }
{3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd +
{\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} \right) }^{\frac{1}{3}}} -
\left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd + {\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} }}
{54}\right)^\frac{1}{3} +
\frac{-a^3 + 4ab - 8c}
{4{\sqrt{\frac{a^2}{4} - \frac{2b}{3} +
\frac{2^{\frac{1}{3}}
\left( b^2 - 3ac + 12d \right) }{3
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd +
{\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 +
27a^2d - 72bd \right) }^2}} \right) }^
{\frac{1}{3}}} +
\left( \frac{{ 2b^3 - 9abc + 27c^2 +
27a^2d - 72bd +
{\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 +
27a^2d - 72bd \right) }^2}} }
}{54}\right)^\frac{1}{3}}}}}}} \\
r_4 & = &
{\frac{-a}{4} + \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} +
\frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) }
{3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd +
{\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} \right) }^{\frac{1}{3}}} +
\left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd + {\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} }}
{54}\right)^\frac{1}{3}}} +
\frac{1}{2}{\sqrt{\frac{a^2}{2} - \frac{4b}{3} -
\frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) }
{3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd +
{\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} \right) }^{\frac{1}{3}}} -
\left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd + {\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd \right) }^2}} }}
{54} \right)^\frac{1}{3} +
\frac{-a^3 + 4ab - 8c}
{4{\sqrt{\frac{a^2}{4} - \frac{2b}{3} +
\frac{2^{\frac{1}{3}}
\left( b^2 - 3ac + 12d \right) }{3
{\left( 2b^3 - 9abc + 27c^2 + 27a^2d -
72bd +
{\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 +
27a^2d - 72bd \right) }^2}} \right) }^
{\frac{1}{3}}} +
\left( \frac{{ 2b^3 - 9abc + 27c^2 +
27a^2d - 72bd +
{\sqrt{-4
{\left( b^2 - 3ac + 12d \right) }^3 +
{\left( 2b^3 - 9abc + 27c^2 +
27a^2d - 72bd \right) }^2}} }
}{54}\right)^\frac{1}{3}}}}}}}
\end{eqnarray*}
The formulas for the roots are much too unwieldy to be used for solving quartic equations by radicals, even with the help of a computer. A practical algorithm for solving quartic equations by radicals is given in the concluding paragraph of the Galois-theoretic derivation of the quartic formula.