This article is about cubic equations in one variable. For cubic equations in
two variables, see
cubic plane curve.
In mathematics, a cubic function is a function of the form
- $f(x)=ax^3+bx^2+cx+d,\backslash ,$
where a is nonzero; or in other words, a function defined by a polynomial of degree three. The derivative of a cubic function is a quadratic function. The integral of a cubic function is a quartic function.
Setting ƒ(x) = 0 produces a cubic equation of the form:
- $ax^3+bx^2+cx+d=0.\backslash ,$
Usually, the coefficients a, b,c, d are real numbers. However, most of the theory is also valid if they belong to any field of characteristic other than 2 or 3.
To solve a cubic equation is to find the roots (zeros) of a cubic function.
There are various ways to solve a cubic equation. The roots of a cubic, like those of a quadratic or quartic (fourth degree) function but no higher degree function (by the Abel–Ruffini theorem), can always be found algebraically (as a formula involving simple functions like the square root and cube root functions). The roots can also be found trigonometrically. Alternatively, one can find a numerical approximation of the roots in the field of the real or complex numbers. This may be obtained by any root-finding algorithm, like Newton's method.
Solving cubic equations is a necessary part of solving the general quartic equation, since solving the latter requires solving its resolvent cubic equation.
History
Cubic equations were known to ancient Greek mathematician Diophantus;^{[1]} even earlier to ancient Babylonians who were able to solve certain cubic equations;^{[2]} and also to the ancient Egyptians. Doubling the cube is the simplest and oldest studied cubic equation, and one which the ancient Egyptians considered to be impossible.^{[3]} Hippocrates reduced this problem to that of finding two mean proportionals between one line and another of twice its length, but could not solve this with a compass and straightedge construction,^{[4]} a task which is now known to be impossible. Hippocrates, Menaechmus and Archimedes are believed to have come close to solving the problem of doubling the cube using intersecting conic sections,^{[4]} though historians such as Reviel Netz dispute whether the Greeks were thinking about cubic equations or just problems that can lead to cubic equations. Some others like T. L. Heath, who translated all Archimedes' works, disagree, putting forward evidence that Archimedes really solved cubic equations using intersections of two cones, but also discussed the conditions where the roots are 0, 1 or 2.^{[5]}
In the 7th century, the Tang dynasty astronomer mathematician Wang Xiaotong in his mathematical treatise titled Jigu Suanjing systematically established and solved 25 cubic equations of the form $x^3+px^2+qx=N$, 23 of them with $p,q\; \backslash ne\; 0$, and two of them with $q\; =\; 0$.^{[6]}
In the 11th century, the Persian poet-mathematician, Omar Khayyám (1048–1131), made significant progress in the theory of cubic equations. In an early paper he wrote regarding cubic equations, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution.^{[7]}^{[8]} In his later work, the Treatise on Demonstration of Problems of Algebra, he wrote a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections.^{[9]}^{[10]}
In the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of a cubic equation:^{[11]}
$x^3+12x=6x^2+35$
In the 12th century, another Persian mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), wrote the Al-Mu'adalat (Treatise on Equations), which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what would later be known as the "Ruffini-Horner method" to numerically approximate the root of a cubic equation. He also developed the concepts of a derivative function and the maxima and minima of curves in order to solve cubic equations which may not have positive solutions.^{[12]} He understood the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations.^{[13]}
Leonardo de Pisa, also known as Fibonacci (1170–1250), was able to find the positive solution to the cubic equation x^{3} + 2x^{2} + 10x = 20, using the Babylonian numerals. He gave the result as 1,22,7,42,33,4,40 (equivalent to 1 + 22/60 + 7/60^{2} + 42/60^{3} + 33/60^{4} + 4/60^{5} + 40/60^{6}),^{[14]} which differs from the correct value by only about three trillionths.
In the early 16th century, the Italian mathematician Scipione del Ferro (1465–1526) found a method for solving a class of cubic equations, namely those of the form x^{3} + mx = n. In fact, all cubic equations can be reduced to this form if we allow m and n to be negative, but negative numbers were not known to him at that time. Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fiore about it.
In 1530, Niccolò Tartaglia (1500–1557) received two problems in cubic equations from Zuanne da Coi and announced that he could solve them. He was soon challenged by Fiore, which led to a famous contest between the two. Each contestant had to put up a certain amount of money and to propose a number of problems for his rival to solve. Whoever solved more problems within 30 days would get all the money. Tartaglia received questions in the form x^{3} + mx = n, for which he had worked out a general method. Fiore received questions in the form x^{3} + mx^{2} = n, which proved to be too difficult for him to solve, and Tartaglia won the contest.
Later, Tartaglia was persuaded by Gerolamo Cardano (1501–1576) to reveal his secret for solving cubic equations. In 1539, Tartaglia did so only on the condition that Cardano would never reveal it and that if he did reveal a book about cubics, that he would give Tartaglia time to publish. Some years later, Cardano learned about Ferro's prior work and published Ferro's method in his book Ars Magna in 1545, meaning Cardano gave Tartaglia 6 years to publish his results (with credit given to Tartaglia for an independent solution). Cardano's promise with Tartaglia stated that he not publish Tartaglia's work, and Cardano felt he was publishing del Ferro's, so as to get around the promise. Nevertheless, this led to a challenge to Cardano by Tartaglia, which Cardano denied. The challenge was eventually accepted by Cardano's student Lodovico Ferrari (1522–1565). Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and income.^{[15]}
Cardano noticed that Tartaglia's method sometimes required him to extract the square root of a negative number. He even included a calculation with these complex numbers in Ars Magna, but he did not really understand it. Rafael Bombelli studied this issue in detail and is therefore often considered as the discoverer of complex numbers.
François Viète (1540–1603) independently derived the trigonometric solution for the cubic with three real roots, and René Descartes (1596–1650) extended the work of Viète.^{[16]}
Derivative
Template:Cubic graph special points.svg
Through the quadratic formula the roots of the derivative f ′(x) = 3ax^{2} + 2bx + c are given by
- $x=\backslash frac\{-b\; \backslash pm\; \backslash sqrt\; \{b^2-3ac\}\}\{3a\}$
and provide the critical points where the slope of the cubic function is zero. If b^{2} − 3ac > 0, then the cubic function has a local maximum and a local minimum. If b^{2} − 3ac = 0, then the cubic's inflection point is the only critical point. If b^{2} − 3ac < 0, then there are no critical points. In the cases where b^{2} − 3ac ≤ 0, the cubic function is strictly monotonic.
Roots of a cubic function
The general cubic equation has the form
- $ax^3+bx^2+cx+d=0\; \backslash qquad(1)$
with $a\backslash neq\; 0\backslash ,.$
This section describes how the roots of such an equation may be computed. The coefficients a, b, c, d are generally assumed to be real numbers, but most of the results apply when they belong to any field of characteristic not 2 or 3.
The nature of the roots
Every cubic equation (1) with real coefficients has at least one solution x among the real numbers; this is a consequence of the intermediate value theorem. We can distinguish several possible cases using the discriminant,
- $\backslash Delta\; =\; 18abcd\; -4b^3d\; +\; b^2c^2\; -\; 4ac^3\; -\; 27a^2d^2.\; \backslash ,$
The following cases need to be considered:
^{[17]}
- If Δ > 0, then the equation has three distinct real roots.
- If Δ = 0, then the equation has a multiple root and all its roots are real.
- If Δ < 0, then the equation has one real root and two nonreal complex conjugate roots.
For information about the location in the complex plane of the roots of a polynomial of any degree, including degree three, see Properties of polynomial roots and Routh–Hurwitz stability criterion
General formula for roots
For the general cubic equation
- $a\; x^3\; +\; b\; x^2\; +\; c\; x\; +\; d\; =\; 0$
the general formula for the roots, in terms of the coefficients, is as follows:^{[18]}^{[19]}
- $x\_k\; =\; -\; \backslash frac\{1\}\{3a\}\backslash left(b\backslash \; +\backslash \; u\_k\; C\backslash \; +\backslash \; \backslash frac\{\backslash Delta\_0\}\{u\_kC\}\backslash right)\backslash \; ,\; \backslash qquad\; k\; \backslash in\; \backslash \{1,2,\; 3\backslash \}$
where
- $u\_1\; =\; 1\backslash \; ,\backslash qquad\; u\_2\; =\; \{-1\; +\; i\backslash sqrt\{3\}\; \backslash over\; 2\}\backslash \; ,\backslash qquad\; u\_3\; =\; \{-1\; -\; i\backslash sqrt\{3\}\; \backslash over\; 2\}$
are the three cubic roots of unity, and where
- $C\; =\; \backslash sqrt[3]\{\backslash frac\{\backslash Delta\_1\; +\; \backslash sqrt\{\backslash Delta\_1^2\; -\; 4\; \backslash Delta\_0^3\}\}\{2\}\}\; \backslash qquad\; \backslash qquad\; \{\backslash color\{white\}.\}$ (see below for special cases)
with
- $\backslash Delta\_0\; =\; b^2-3\; a\; c$
- $\backslash Delta\_1\; =\; 2\; b^3-9\; a\; b\; c+27\; a^2\; d$
and
- $\backslash Delta\_1^2\; -\; 4\; \backslash Delta\_0^3\; =\; -27\backslash ,a^2\backslash ,\backslash Delta\backslash \; ,$ where $\backslash Delta$ is the discriminant discussed above.
In these formulae, $\backslash sqrt\{~~\}$ and $\backslash sqrt[3]\{~~\}$ denote any choice for the square or cubic roots. Changing of choice for the square root amounts to exchanging $x\_2$ and $x\_3$. Changing of choice for the cubic root amounts to circularly permute the roots. Thus the freeness of choosing a determination of the square or cubic roots corresponds exactly to the freeness for numbering the roots of the equation.
Four centuries ago, Gerolamo Cardano proposed a similar formula (see below), which appears yet in many textbooks:
- $x\_k\; =\; -\; \backslash frac\{1\}\{3a\}(b\backslash \; +\backslash \; u\_k\; C\backslash \; +\backslash \; \backslash bar\; u\_k\; \backslash bar\; C)$
where
- $\backslash bar\; C\; =\; \backslash sqrt[3]\{\backslash frac\{\backslash Delta\_1\; -\; \backslash sqrt\{\backslash Delta\_1^2\; -\; 4\; \backslash Delta\_0^3\}\}\{2\}\}$
and $\backslash bar\; u\_k$ is the complex conjugate of $u\_k$ (note that $C\backslash bar\; C=\backslash Delta\_0$).
However, this formula is applicable without further explanation only when a, b, c, d are real numbers and the operand of the square root $\backslash Delta\_1^2\; -\; 4\; \backslash Delta\_0^3\; \backslash ge\; 0$ is non-negative. When this operand is real and non-negative, the square root refers to the principal (positive) square root and the cube roots in the formula are to be interpreted as the real ones. Otherwise, there is no real square root and one can arbitrarily choose one of the imaginary square roots (the same one everywhere in the solution). For extracting the complex cube roots of the resulting complex expression, we have also to choose among three cube roots in each part of each solution, giving nine possible combinations of one of three cube roots for the first part of the expression and one of three for the second. The correct combination is such that the two cube roots chosen for the two terms in a given solution expression are complex conjugates of each other (whereby the two imaginary terms in each solution cancel out).
The next sections describe how these formulas may be obtained.
Special cases
If $\backslash Delta\; \backslash neq\; 0$ and $\backslash Delta\_0\; =\; 0,$ the sign of $\backslash sqrt\{\backslash Delta\_1^2\; -\; 4\; \backslash Delta\_0^3\}=\backslash sqrt\{\backslash Delta\_1^2\}$ has to be chosen to have $C\; \backslash neq\; 0,$ that is one should define $\backslash sqrt\{\backslash Delta\_1^2\}\; =\backslash Delta\_1,$ whichever is the sign of $\backslash Delta\_1.$
If $\backslash Delta\; =\; 0$ and $\backslash \; \backslash Delta\_0\; =\; 0,$ the three roots are equal:
- $x\_1=x\_2=x\_3=-\backslash frac\{b\}\{3a\}.$
If $\backslash Delta=0$ and $\backslash Delta\_0\; \backslash neq\; 0,$ the above expression for the roots is correct but misleading, hiding the fact that no radical is needed to represent the roots. In fact, in this case, there is a double root,
- $x\_1=x\_2=\backslash frac\{9ad-bc\}\{2\backslash Delta\_0\},$
and a simple root
- $x\_3=\backslash frac\{4abc-9a^2d-b^3\}\{a\backslash Delta\_0\}.$
Reduction to a depressed cubic
Dividing Equation (1) by $a$ and substituting $x$ by $t-\backslash frac\{b\}\{3a\}$ (the Tschirnhaus transformation) we get the equation
- $t^3+pt+q=0\; \backslash qquad(2)$
where
- $\backslash begin\{align\}$
p=&\frac{3ac-b^2}{3a^2}\\
q=&\frac{2b^3-9abc+27a^2d}{27a^3}.
\end{align}
The left hand side of equation (2) is a monic trinomial called a depressed cubic.
Any formula for the roots of a depressed cubic may be transformed into a formula for the roots of Equation (1) by substituting the above values for $p$ and $q$ and using the relation $x=t-\backslash frac\{b\}\{3a\}$.
Therefore, only Equation (2) is considered in the following.
Cardano's method
The solutions can be found with the following method due to Scipione del Ferro and Tartaglia, published by Gerolamo Cardano in 1545.^{[20]}
This method applies to the depressed cubic
- $t^3\; +\; pt\; +\; q\; =\; 0\backslash ,.\; \backslash qquad\; (2)$
We introduce two variables u and v linked by the condition
- $u+v=t\backslash ,$
and substitute this in the depressed cubic (2), giving
- $u^3+v^3+(3uv+p)(u+v)+q=0\; \backslash qquad\; (3)\backslash ,$.
At this point Cardano imposed a second condition for the variables u and v:
- $3uv+p=0\backslash ,$.
As the first parenthesis vanishes in (3), we get $u^3+v^3=-q$ and $u^3v^3=-p^3/27$. Thus $u^3$ and $v^3$ are the two roots of the equation
- $z^2\; +\; qz\; -\; \{p^3\backslash over\; 27\}\; =\; 0\backslash ,.$
At this point, Cardano, who did not know complex numbers, supposed that the roots of this equation were real, that is that $\backslash frac\{q^2\}\{4\}+\backslash frac\{p^3\}\{27\}\; >0\backslash ,.$
Solving this equation and using the fact that $u$ and $v$ may be exchanged, we find
- $u^\{3\}=-\{q\backslash over\; 2\}\; +\; \backslash sqrt$ and $v^\{3\}=-\{q\backslash over\; 2\}\; -\; \backslash sqrt$.
As these expressions are real, their cube roots are well defined and, like Cardano, we get
- $t\_1=u+v=\backslash sqrt[3]\{-\{q\backslash over\; 2\}+\; \backslash sqrt\}\; +\backslash sqrt[3]\{-\{q\backslash over\; 2\}-\; \backslash sqrt\}$
The two complex roots are obtained by considering the complex cubic roots; the fact $uv$ is real implies that they are obtained by multiplying one of the above cubic roots by $\backslash ,\backslash tfrac\{-1\}\{2\}\; +\; i\backslash tfrac\{\backslash sqrt\{3\}\}\{2\}\backslash ,$ and the other by $\backslash ,\backslash tfrac\{-1\}\{2\}\; -\; i\backslash tfrac\{\backslash sqrt\{3\}\}\{2\}\backslash ,$.
If $\backslash frac\{q^2\}\{4\}+\backslash frac\{p^3\}\{27\}\backslash ,$ is not necessarily positive, we have to choose a cube root of $u^3$. As there is no direct way to choose the corresponding cube root of $v^3$, one has to use the relation $v=-\backslash frac\{p\}\{3u\}$, which gives
- $u=\backslash sqrt[3]\{-\{q\backslash over\; 2\}-\; \backslash sqrt\}\; \backslash qquad\; (4)$
and
- $t=u-\backslash frac\{p\}\{3u\}\backslash ,.$
Note that the sign of the square root does not affect the resulting $t$, because changing it amounts to exchanging $u$ and $v$. We have chosen the minus sign to have $u\backslash ne\; 0$ when $p\; =\; 0$ and $q\backslash ne\; 0$, in order to avoid a division by zero. With this choice, the above expression for $t$ always works, except when $p\; =\; q=0$, where the second term becomes 0/0. In this case there is a triple root $t=0$.
Note also that in several cases the solutions are expressed with fewer square or cube roots
- If $p=q=0$ then we have the triple real root
- $t=0.\backslash ,$
- If $p=0$ and $q\backslash ne\; 0$ then
- $u=-\backslash sqrt[3]\{q\}\; \backslash text\{\; and\; \}\; v\; =\; 0$
- and the three roots are the three cube roots of $-q$.
- If $p\backslash ne\; 0$ and $q=0$ then
- $u=\backslash sqrt$Template:P\over 3 \qquad \text{and} \qquad v=-\sqrtTemplate:P\over 3,
- in which case the three roots are
- $t=u+v=0\; ,\; \backslash qquad\; t=\backslash omega\_1u-\{p\backslash over\; 3\backslash omega\_1u\}=\backslash sqrt\{-p\}\; ,\; \backslash qquad\; t=\{u\backslash over\; \backslash omega\_1\}-\{\backslash omega\_1p\backslash over\; 3u\}=-\backslash sqrt\{-p\}\; ,$
- where
- $\backslash omega\_1=e^\{i\backslash frac\{2\backslash pi\}\{3\}\}=-\backslash tfrac\{1\}\{2\}\; +\; \backslash tfrac\{\backslash sqrt\{3\}\}\{2\}i.$
- Finally if $4p^3+27q^2=0\; \backslash text\{\; and\; \}\; p\backslash ne\; 0$, there is a double root and a simple root which may be expressed rationally in term of $p\; \backslash text\{\; and\; \}\; q$, but this expression may not be immediately deduced from the general expression of the roots:
- $t\_1=t\_2=\; -\backslash frac\{3q\}\{2p\}\backslash quad\; \backslash text\{and\}\; \backslash quad\; t\_3=\backslash frac\{3q\}\{p\}\backslash ,.$
To pass from these roots of $t$ in Equation (2) to the general formulas for roots of $x$ in Equation (1), subtract $\backslash frac\{b\}\{3a\}$ and replace $p$ and $q$ by their expressions in terms of $a,b,c,d$.
Vieta's substitution
Starting from the depressed cubic
- $t^3\; +\; pt\; +\; q\; =\; 0,$
we make the following substitution, known as Vieta's substitution:
- $t\; =\; w\; -\; \backslash frac\{p\}\{3w\}$
This results in the equation
- $w^3\; +\; q\; -\; \backslash frac\{p^3\}\{27w^3\}\; =\; 0.$
Multiplying by w^{3}, it becomes a sextic equation in w, which is in fact a quadratic equation in w^{3}:
- $w^6\; +\; qw^3\; -\; \backslash frac\{p^3\}\{27\}\; =\; 0$
The quadratic formula allows to solve it in w^{3}. If w_{1}, w_{2} and w_{3} are the three cubic roots of one of the solutions in w^{3}, then the roots of the original depressed cubic are
- $t\_1\; =\; w\_1\; -\; \backslash frac\{p\}\{3w\_1\},\; \backslash quad\; t\_2\; =\; w\_2\; -\; \backslash frac\{p\}\{3w\_2\}\backslash quad\backslash text\{and\}\; \backslash quad\; t\_3\; =\; w\_3\; -\; \backslash frac\{p\}\{3w\_3\}.$
Lagrange's method
In his paper Réflexions sur la résolution algébrique des équations ("Thoughts on the algebraic solving of equations"), Joseph Louis Lagrange introduced a new method to solve equations of low degree.
This method works well for cubic and quartic equations, but Lagrange did not succeed in applying it to a quintic equation, because it requires solving a resolvent polynomial of degree at least six.^{[21]}^{[22]}^{[23]} This is explained by the Abel–Ruffini theorem, which proves that such polynomials cannot be solved by radicals. Nevertheless the modern methods for solving solvable quintic equations are mainly based on Lagrange's method.^{[23]}
In the case of cubic equations, Lagrange's method gives the same solution as Cardano's, where the latter may seem almost magical to the modern reader. But Cardano explains in his book Ars Magna how he arrived at the idea of considering the unknown of the cubic equation as a sum of two other quantities, by drawing attention to a geometrical problem that involves two cubes of different size. Lagrange's method may also be applied directly to the general cubic equation (1) without using the reduction to the depressed cubic equation (2). Nevertheless the computation is much easier with this reduced equation.
Suppose that x_{0}, x_{1} and x_{2} are the roots of equation (1) or (2), and define $\backslash zeta\; =\; -\backslash tfrac\{1\}\{2\}\; +\; \backslash tfrac\{\backslash sqrt\{3\}\}\{2\}i$, so that ζ is a primitive third root of unity which satisfies the relation $\backslash zeta^2+\backslash zeta+1=0$. We now set
- $s\_0\; =\; x\_0\; +\; x\_1\; +\; x\_2,\backslash ,$
- $s\_1\; =\; x\_0\; +\; \backslash zeta\; x\_1\; +\; \backslash zeta^2\; x\_2,\backslash ,$
- $s\_2\; =\; x\_0\; +\; \backslash zeta^2\; x\_1\; +\; \backslash zeta\; x\_2.\backslash ,$
This is the discrete Fourier transform of the roots: observe that while the coefficients of the polynomial are symmetric in the roots, in this formula an order has been chosen on the roots, so these are not symmetric in the roots.
The roots may then be recovered from the three s_{i} by inverting the above linear transformation via the inverse discrete Fourier transform, giving
- $x\_0\; =\; \backslash tfrac13(s\_0\; +\; s\_1\; +\; s\_2),\backslash ,$
- $x\_1\; =\; \backslash tfrac13(s\_0\; +\; \backslash zeta^2\; s\_1\; +\; \backslash zeta\; s\_2),\backslash ,$
- $x\_2\; =\; \backslash tfrac13(s\_0\; +\; \backslash zeta\; s\_1\; +\; \backslash zeta^2\; s\_2).\backslash ,$
The polynomial $s\_0$ is an elementary symmetric polynomial and is thus equal to $-b/a$ in case of Equation (1) and to zero in case of Equation (2), so we only need to seek values for the other two.
The polynomials $s\_1$ and $s\_2$ are not symmetric functions of the roots: $s\_0$ is invariant, while the two non-trivial cyclic permutations of the roots send $s\_1$ to $\backslash zeta\; s\_1$ and $s\_2$ to $\backslash zeta^2\; s\_2$, or $s\_1$ to $\backslash zeta^2\; s\_1$ and $s\_2$ to $\backslash zeta\; s\_2$ (depending on which permutation), while transposing $x\_1$ and $x\_2$ switches $s\_1$ and $s\_2$; other transpositions switch these roots and multiply them by a power of $\backslash zeta.$
Thus, $s\_1^3$, $s\_2^3$ and $s\_1\; s\_2$ are left invariant by the cyclic permutations of the roots, which multiply them by $\backslash zeta^3=1$. Also $s\_1\; s\_2$ and $s\_1^3+s\_2^3$ are left invariant by the transposition of $x\_1$ and $x\_2$ which exchanges $s\_1$ and $s\_2$. As the permutation group $S\_3$ of the roots is generated by these permutations, it follows that $s\_1^3+s\_2^3$ and $s\_1\; s\_2$ are symmetric functions of the roots and may thus be written as polynomials in the elementary symmetric polynomials and thus as rational functions of the coefficients of the equation. Let $s\_1^3+s\_2^3=A$ and $s\_1\; s\_2=B$ in these expressions, which will be explicitly computed below.
We have that $s\_1^3$ and $s\_2^3$ are the two roots of the quadratic equation
- $z^2-Az+B^3\; =\; 0\; \backslash ,.$
Thus the resolution of the equation may be finished exactly as described for Cardano's method, with $s\_1$ and $s\_2$ in place of $u$ and $v$.
Computation of A and B
Setting $E\_1=x\_0+x\_1+x\_2$, $E\_2=x\_0x\_1+x\_1x\_2+x\_2x\_0$ and $E\_3=x\_0x\_1x\_2$, the elementary symmetric polynomials, we have, using that $\backslash zeta^3=1$:
- $s\_1^3=x\_0^3+x\_1^3+x\_2^3+3\backslash zeta\; (x\_0^2x\_1+x\_1^2x\_2+x\_2^2x\_0)\; +3\backslash zeta^2\; (x\_0x\_1^2+x\_1x\_2^2+x\_2x\_0^2)\; +6x\_0x\_1x\_2\backslash ,.$
The expression for $s\_2^3$ is the same with $\backslash zeta$ and $\backslash zeta^2$ exchanged. Thus, using $\backslash zeta^2+\backslash zeta=-1$ we get
- $$
A=s_1^3+s_2^3=2(x_0^3+x_1^3+x_2^3)-3(x_0^2x_1+x_1^2x_2+x_2^2x_0+x_0x_1^2+x_1x_2^2+x_2x_0^2)+12x_0x_1x_2\,,
and a straightforward computation gives
- $$
A=s_1^3+s_2^3=2E_1^3-9E_1E_2+27E_3\,.
Similarly we have
- $$
B=s_1s_2=x_0^2+x_1^2+x_2^2+(\zeta+\zeta^2)(x_0x_1+x_1x_2+x_2x_0)=E_1^2-3E_2\,.
When solving Equation (1) we have
- $E\_1=-b/a$, $E\_2=c/a$ and $E\_3=-d/a$
With Equation (2), we have $E\_1=0$, $E\_2=p$ and $E\_3=-q$ and thus:
- $A=-27q$ and $B=-3p$.
Note that with Equation (2), we have $x\_0\; =\; \backslash tfrac13(s\_1\; +\; s\_2)$ and $s\_1s\_2=-3p$, while in Cardano's method we have set $x\_0\; =\; u+v$ and $uv=-\backslash frac13p\backslash ,.$
Thus we have, up to the exchange of $u$ and $v$:
- $s\_1=3u$ and $s\_2=3v$.
In other words, in this case, Cardano's and Lagrange's method compute exactly the same things, up to a factor of three in the auxiliary variables, the main difference being that Lagrange's method explains why these auxiliary variables appear in the problem.
Trigonometric (and hyperbolic) method
When a cubic equation has three real roots, the formulas expressing these roots in terms of radicals involve complex numbers. It has been proved that when none of the three real roots is rational—the casus irreducibilis— one cannot express the roots in terms of real radicals. Nevertheless, purely real expressions of the solutions may be obtained using hypergeometric functions,^{[24]} or more elementarily in terms of trigonometric functions, specifically in terms of the cosine and arccosine functions.
The formulas which follow, due to François Viète,^{[16]} are true in general (except when p = 0), are purely real when the equation has three real roots, but involve complex cosines and arccosines when there is only one real root.
Starting from Equation (2), $t^3+pt+q=0$, let us set $t=u\backslash cos\backslash theta\backslash ,.$ The idea is to choose $u$ to make Equation (2) coincide with the identity
- $4\backslash cos^3\backslash theta-3\backslash cos\backslash theta-\backslash cos(3\backslash theta)=0\backslash ,.$
In fact, choosing $u=2\backslash sqrt\{-\backslash frac\{p\}\{3\}\}$ and dividing Equation (2) by $\backslash frac\{u^3\}\{4\}$ we get
- $4\backslash cos^3\backslash theta-3\backslash cos\backslash theta-\backslash frac\{3q\}\{2p\}\backslash sqrt\{\backslash frac\{-3\}\{p\}\}=0\backslash ,.$
Combining with the above identity, we get
- $\backslash cos(3\backslash theta)=\backslash frac\{3q\}\{2p\}\backslash sqrt\{\backslash frac\{-3\}\{p\}\}$
and thus the roots are^{[25]}
- $t\_k=2\backslash sqrt\{-\backslash frac\{p\}\{3\}\}\backslash cos\backslash left(\backslash frac\{1\}\{3\}\backslash arccos\backslash left(\backslash frac\{3q\}\{2p\}\backslash sqrt\{\backslash frac\{-3\}\{p\}\}\backslash right)-k\backslash frac\{2\backslash pi\}\{3\}\backslash right)\; \backslash quad\; \backslash text\{for\}\; \backslash quad\; k=0,1,2\; \backslash ,.$
This formula involves only real terms if $p<0$ and the argument of the arccosine is between −1 and 1. The last condition is equivalent to $4p^3+27q^2\backslash leq\; 0\backslash ,,$ which implies also $p<0$. Thus the above formula for the roots involves only real terms if and only if the three roots are real.
Denoting by $C(p,q)$ the above value of t_{0}, and using the inequality $-\backslash pi\backslash le\; \backslash arccos(u)\; \backslash le\; \backslash pi$ for a real number u such that $-1\backslash le\; u\backslash le\; 1\backslash ,,$ the three roots may also be expressed as
- $t\_0=C(p,q),\backslash qquad\; t\_2=-C(p,-q),\; \backslash qquad\; t\_1=-t\_0-t\_2\backslash ,.$
If the three roots are real, we have
- $t\_0\backslash ge\; t\_1\backslash ge\; t\_2\backslash ,.$
All these formulas may be straightforwardly transformed into formulas for the roots of the general cubic equation (1), using the back substitution described in Section Reduction to a depressed cubic.
When there is only one real root (and p ≠ 0), it may be similarly represented using hyperbolic functions, as^{[26]}^{[27]}
- $t\_0=-2\backslash frac\{|q|\}\{q\}\backslash sqrt\{-\backslash frac\{p\}\{3\}\}\backslash cosh\backslash left(\backslash frac\{1\}\{3\}\backslash operatorname\{arcosh\}\backslash left(\backslash frac\{-3|q|\}\{2p\}\backslash sqrt\{\backslash frac\{-3\}\{p\}\}\backslash right)\backslash right)\; \backslash quad\; \backslash text\{if\; \}\; \backslash quad\; 4p^3+27q^2>0\; \backslash text\{\; and\; \}\; p<0\backslash ,,$
- $t\_0=-2\backslash sqrt\{\backslash frac\{p\}\{3\}\}\backslash sinh\backslash left(\backslash frac\{1\}\{3\}\backslash operatorname\{arsinh\}\backslash left(\backslash frac\{3q\}\{2p\}\backslash sqrt\{\backslash frac\{3\}\{p\}\}\backslash right)\backslash right)\; \backslash quad\; \backslash text\{if\; \}\; \backslash quad\; p>0\backslash ,.$
If p ≠ 0 and the inequalities on the right are not satisfied the formulas remain valid but involve complex quantities.
When $p=\backslash pm\; 3$, the above values of $t\_0$ are sometimes called the Chebyshev cube root.^{[28]} More precisely, the values involving cosines and hyperbolic cosines define, when $p=-3$, the same analytic function denoted $C\_\{\backslash frac13\}(q)$, which is the proper Chebyshev cube root. The value involving hyperbolic sines is similarly denoted $S\_\{\backslash frac13\}(q),$ when $p=3$.
Factorization
If the cubic equation $ax^3\; +\; bx^2\; +\; cx\; +d=0$ with integer coefficients has a rational real root, it can be found using the rational root test: If the root is r = m / n fully reduced, then m is a factor of d and n is a factor of a, so all possible combinations of values for m and n can be checked for whether they satisfy the cubic equation.
The rational root test may also be used for a cubic equation with rational coefficients: by multiplication by the lowest common denominator) of the coefficients, one gets an equation with integer coefficients which has exactly the same roots.
The rational root test is particularly useful when there are three real roots because the algebraic solution unhelpfully expresses the real roots in terms of complex entities. The rational root test is also helpful in the presence of one real and two complex roots because it allows all of the roots to be written without the use of cube roots.
If r is any root of the cubic, then we may factor out (x–r ) using polynomial long division to obtain
- $\backslash left\; (x-r\backslash right\; )\backslash left\; (ax^2+(b+ar)x+c+br+ar^2\; \backslash right\; )\; =\; ax^3+bx^2+cx+d\backslash ,.$
Hence if we know one root we can find the other two by using the quadratic formula to solve the quadratic $ax^2+(b+ar)x+c+br+ar^2$, giving
- $\backslash frac\{-b-ra\; \backslash pm\; \backslash sqrt\{b^2-4ac-2abr-3a^2r^2\}\}\{2a\}$
for the other two roots.
Geometric interpretation of the roots
Three real roots
Viète's trigonometric expression of the roots in the three-real-roots case lends itself to a geometric interpretation in terms of a circle.^{[16]}^{[29]} When the cubic is written in depressed form as above as $t^3+pt+q=0$, as shown above the solution can be expressed as
- $t\_k=2\backslash sqrt\{-\backslash frac\{p\}\{3\}\}\backslash cos\backslash left(\backslash frac\{1\}\{3\}\backslash arccos\backslash left(\backslash frac\{3q\}\{2p\}\backslash sqrt\{\backslash frac\{-3\}\{p\}\}\backslash right)-k\backslash frac\{2\backslash pi\}\{3\}\backslash right)\; \backslash quad\; \backslash text\{for\}\; \backslash quad\; k=0,1,2\; \backslash ,.$
Here $\backslash arccos\backslash left(\backslash frac\{3q\}\{2p\}\backslash sqrt\{\backslash frac\{-3\}\{p\}\}\backslash right)$ is an angle in the unit circle; taking $\backslash tfrac\{1\}\{3\}$ of that angle corresponds to taking a cube root of a complex number; adding $-k\backslash frac\{2\backslash pi\}\{3\}$ for k = 1, 2 finds the other cube roots; and multiplying the cosines of these resulting angles by $2\backslash sqrt\{-\backslash frac\{p\}\{3\}\}$ corrects for scale.
For the non-depressed case $x^3+bx^2+cx+d=0$ (shown in the accompanying graph), the depressed case as indicated previously is obtained by defining t such that $x=t-\backslash tfrac\{b\}\{3\}$ so $t=x+\backslash tfrac\{b\}\{3\}$. Graphically this corresponds to simply shifting the graph horizontally when changing between the variables t and x, without changing the angle relationships.
One real and two complex roots
In the Cartesian plane
If a cubic is plotted in the Cartesian plane, the real root can be seen graphically as the horizontal intercept of the curve. But further,^{[30]}^{[31]}^{[32]} if the complex conjugate roots are written as g+hi, then g is the abscissa (the positive or negative horizontal distance from the origin) of the tangency point of a line that is tangent to the cubic curve and intersects the horizontal axis at the same place as does the cubic curve; and |h| is the square root of the tangent of the angle between this line and the horizontal axis.
In the complex plane
With one real and two complex roots, the three roots can be represented as points in the complex plane, as can the two roots of the cubic's derivative. There is an interesting geometrical relationship among all these roots.
The points in the complex plane representing the three roots serve as the vertices of an isosceles triangle. (The triangle is isosceles because one root is on the horizontal (real) axis and the other two roots, being complex conjugates, appear symmetrically above and below the real axis.) Marden's Theorem says that the points representing the roots of the derivative of the cubic are the foci of the Steiner inellipse of the triangle—the unique ellipse that is tangent to the triangle at the midpoints of its sides. If the angle at the vertex on the real axis is less than $\backslash tfrac\{\backslash pi\}\{3\}$ then the major axis of the ellipse lies on the real axis, as do its foci and hence the roots of the derivative. If that angle is greater than $\backslash tfrac\{\backslash pi\}\{3\}$, the major axis is vertical and its foci, the roots of the derivative, are complex. And if that angle is $\backslash tfrac\{\backslash pi\}\{3\}$, the triangle is equilateral, the Steiner inellipse is simply the triangle's incircle, its foci coincide with each other at the incenter, which lies on the real axis, and hence the derivative has duplicate real roots.
Omar Khayyám's solution
As shown in this graph, to solve the third-degree equation $x^3\; +\; a^2x\; =\; b$ where $b>0,$ Omar Khayyám constructed the parabola $y=x^2/a,$ the circle with diameter $b/a^2$ having its center on the positive x-axis and intersecting the origin, and a vertical line through the point above the x-axis where the circle and parabola intersect. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the x-axis.
See also
Notes
References
External links
- Template:Springer
- Solving a Cubic by means of Moebius transforms
- Interesting derivation of trigonometric cubic solution with 3 real roots
- Calculator for solving Cubics (also solves Quartics and Quadratics)
- Convergence
- Cubic Equation Solver.
- MacTutor archive.
- Template:Planetmath reference
- Cardano solution calculator as java applet at some local site. Only takes natural coefficients.
- Graphic explorer for cubic functions With interactive animation, slider controls for coefficients
- On Solution of Cubic Equations at Holistic Numerical Methods Institute
- Dave Auckly, Solving the quartic with a pencil American Math Monthly 114:1 (2007) 29—39
- The Wolfram Demonstrations Project, 2007.
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