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A formula for finding the roots of the general cubic equation over the field of complex numbers
+
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c0203501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
A formula for finding the roots of the general cubic equation over the
 +
field of complex numbers
 +
$$x^3 + px + q = 0.\label{1}$$
 +
Any cubic equation can be reduced to
 +
the above form. The Cardano formula for the roots of (1) has the form:
  
Any cubic equation can be reduced to the above form. The Cardano formula for the roots of (1) has the form:
+
$\def\radix#1#2{{}^{{}^{\scriptstyle{#1}}}\kern-6pt\sqrt{#2}}$
 +
$$x= \radix{3}{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}
 +
+ \radix{3}{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}.$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c0203502.png" /></td> </tr></table>
+
In this formula one must choose, for each of the three values of
 +
the cube root
 +
$$\alpha = \radix{3}{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}},$$
 +
that value of the cube root
 +
$$\beta = \radix{3}{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}$$
 +
for which the
 +
relation $\alpha\beta = -p/3$ holds (such a value of $\beta$ always exists). In the Cardano
 +
formula, $p$ and $q$ are arbitrary complex numbers. In the case of
 +
real coefficients $p$ and $q$, the property of the roots being real or
 +
imaginary depends on the sign of the discriminant of the equation,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c0203503.png" /></td> </tr></table>
+
$$D= -27q^2 -4 p^3 = -108\Big(\frac{q^2}{4} + \frac{p^3}{27}\Big).$$
 +
When $D>0$ all three roots are real and distinct. However,
 +
according to Cardano's formula, the roots are expressed in terms of
 +
cube roots of imaginary quantities. Although in this case both the
 +
coefficients and the roots are real, the roots cannot be expressed in
 +
terms of the coefficients by means of radicals of real numbers; for
 +
this reason, the above case is called irreducible. When $D=0$, all roots
 +
are real; when $p$ and $q$ are both non-zero, there is one double and
 +
one single root; and when $p$ and $q$ are both zero, there is one
 +
triple root. When $D<0$, all three roots are distinct, one of them being
 +
a real number and the other two — conjugate complex numbers.
  
In this formula one must choose, for each of the three values of the cube root
+
The Cardano formula is named after G. Cardano, who was the first to
 
+
publish it in 1545.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c0203504.png" /></td> </tr></table>
 
 
 
that value of the cube root
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c0203505.png" /></td> </tr></table>
 
 
 
for which the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c0203506.png" /> holds (such a value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c0203507.png" /> always exists). In the Cardano formula, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c0203508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c0203509.png" /> are arbitrary complex numbers. In the case of real coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c02035010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c02035011.png" />, the property of the roots being real or imaginary depends on the sign of the discriminant of the equation,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c02035012.png" /></td> </tr></table>
 
 
 
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c02035013.png" /> all three roots are real and distinct. However, according to Cardano's formula, the roots are expressed in terms of cube roots of imaginary quantities. Although in this case both the coefficients and the roots are real, the roots cannot be expressed in terms of the coefficients by means of radicals of real numbers; for this reason, the above case is called irreducible. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c02035014.png" />, all roots are real; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c02035015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c02035016.png" /> are both non-zero, there is one double and one single root; and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c02035017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c02035018.png" /> are both zero, there is one triple root. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020350/c02035019.png" />, all three roots are distinct, one of them being a real number and the other two — conjugate complex numbers.
 
 
 
The Cardano formula is named after G. Cardano, who was the first to publish it in 1545.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh,   "Higher algebra" , MIR (1972) (Translated from Russian)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD>
 +
<TD valign="top"> A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)</TD>
 +
</TR></table>
  
  
Line 34: Line 48:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B.L. van der Waerden,   "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD>
 +
<TD valign="top"> B.L. van der Waerden, "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German)</TD>
 +
</TR></table>

Latest revision as of 06:45, 18 December 2012


A formula for finding the roots of the general cubic equation over the field of complex numbers $$x^3 + px + q = 0.\label{1}$$ Any cubic equation can be reduced to the above form. The Cardano formula for the roots of (1) has the form:

$\def\radix#1#2{{}^{{}^{\scriptstyle{#1}}}\kern-6pt\sqrt{#2}}$ $$x= \radix{3}{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}} + \radix{3}{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}.$$

In this formula one must choose, for each of the three values of the cube root $$\alpha = \radix{3}{-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}},$$ that value of the cube root $$\beta = \radix{3}{-\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}$$ for which the relation $\alpha\beta = -p/3$ holds (such a value of $\beta$ always exists). In the Cardano formula, $p$ and $q$ are arbitrary complex numbers. In the case of real coefficients $p$ and $q$, the property of the roots being real or imaginary depends on the sign of the discriminant of the equation,

$$D= -27q^2 -4 p^3 = -108\Big(\frac{q^2}{4} + \frac{p^3}{27}\Big).$$ When $D>0$ all three roots are real and distinct. However, according to Cardano's formula, the roots are expressed in terms of cube roots of imaginary quantities. Although in this case both the coefficients and the roots are real, the roots cannot be expressed in terms of the coefficients by means of radicals of real numbers; for this reason, the above case is called irreducible. When $D=0$, all roots are real; when $p$ and $q$ are both non-zero, there is one double and one single root; and when $p$ and $q$ are both zero, there is one triple root. When $D<0$, all three roots are distinct, one of them being a real number and the other two — conjugate complex numbers.

The Cardano formula is named after G. Cardano, who was the first to publish it in 1545.

References

[1] A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)


Comments

References

[a1] B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
How to Cite This Entry:
Cardano formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cardano_formula&oldid=14215
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article