Cubic equation
In algebra, a cubic equation in one variable is an equation of the form
in which a is nonzero.
The solutions of this equation are called
- arithmetic operations, square roots and cube roots. (This is also true of quadratic (second-degree) and quartic (fourth-degree) equations, but not for higher-degree equations, by the Abel–Ruffini theorem.)
- trigonometrically
- root-finding algorithms such as Newton's method.
The coefficients do not need to be real numbers. Much of what is covered below is valid for coefficients in any field with characteristic other than 2 and 3. The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are irrational (and even non-real) complex numbers.
History
Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, and Egyptians.
In the 3rd century AD, the
In the 7th century, the Tang dynasty astronomer mathematician Wang Xiaotong in his mathematical treatise titled Jigu Suanjing systematically established and solved numerically 25 cubic equations of the form x3 + px2 + qx = N, 23 of them with p, q ≠ 0, and two of them with q = 0.[11]
In the 11th century, the Persian poet-mathematician, Omar Khayyam (1048–1131), made significant progress in the theory of cubic equations. In an early paper, 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.[12][a] 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.[13][14] Khayyam made an attempt to come up with an algebraic formula for extracting cubic roots. He wrote:
“We have tried to express these roots by algebra but have failed. It may be, however, that men who come after us will succeed.”[15]
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: x3 + 12x = 6x2 + 35.
In his book Flos, Leonardo de Pisa, also known as
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 x3 + mx = n. In fact, all cubic equations can be reduced to this form if one allows 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 Fior about it.
In 1535,
Later, Tartaglia was persuaded by
Cardano's promise to Tartaglia said that he would 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 from 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 his income.[20]
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[21] 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.[22]
Factorization
If the coefficients of a cubic equation are
with integer coefficients, is said to be
Finding the roots of a reducible cubic equation is easier than solving the general case. In fact, if the equation is reducible, one of the factors must have degree one, and thus have the form
with q and p being
Thus, one root is and the other roots are the roots of the other factor, which can be found by polynomial long division. This other factor is
(The coefficients seem not to be integers, but must be integers if p / q is a root.)
Then, the other roots are the roots of this
Depressed cubic
Cubics of the form
are said to be depressed. They are much simpler than general cubics, but are fundamental, because the study of any cubic may be reduced by a simple
Let
be a cubic equation. The change of variable
gives a cubic (in t) that has no term in t2.
After dividing by a one gets the depressed cubic equation
with
The roots of the original equation are related to the roots of the depressed equation by the relations
Discriminant and nature of the roots
The nature (real or not, distinct or not) of the roots of a cubic can be determined without computing them explicitly, by using the discriminant.
Discriminant
The
If r1, r2, r3 are the three
The discriminant of the depressed cubic is
The discriminant of the general cubic is
It is the product of and the discriminant of the corresponding depressed cubic. Using the formula relating the general cubic and the associated depressed cubic, this implies that the discriminant of the general cubic can be written as
It follows that one of these two discriminants is zero if and only if the other is also zero, and, if the coefficients are real, the two discriminants have the same sign. In summary, the same information can be deduced from either one of these two discriminants.
To prove the preceding formulas, one can use Vieta's formulas to express everything as polynomials in r1, r2, r3, and a. The proof then results in the verification of the equality of two polynomials.
Nature of the roots
If the coefficients of a polynomial are real numbers, and its discriminant is not zero, there are two cases:
- If the cubic has three distinct real roots
- If the cubic has one real root and two non-real complex conjugate roots.
This can be proved as follows. First, if r is a root of a polynomial with real coefficients, then its complex conjugate is also a root. So the non-real roots, if any, occur as pairs of complex conjugate roots. As a cubic polynomial has three roots (not necessarily distinct) by the fundamental theorem of algebra, at least one root must be real.
As stated above, if r1, r2, r3 are the three roots of the cubic , then the discriminant is
If the three roots are real and distinct, the discriminant is a product of positive reals, that is
If only one root, say r1, is real, then r2 and r3 are complex conjugates, which implies that r2 – r3 is a
Multiple root
If the discriminant of a cubic is zero, the cubic has a multiple root. If furthermore its coefficients are real, then all of its roots are real.
The discriminant of the depressed cubic is zero if If p is also zero, then p = q = 0 , and 0 is a triple root of the cubic. If and p ≠ 0 , then the cubic has a simple root
and a double root
In other words,
This result can be proved by expanding the latter product or retrieved by solving the rather simple system of equations resulting from Vieta's formulas.
By using the reduction of a depressed cubic, these results can be extended to the general cubic. This gives: If the discriminant of the cubic is zero, then
- either, if the cubic has a triple root
- and
- or, if the cubic has a double root
- and a simple root,
- and thus
Characteristic 2 and 3
The above results are valid when the coefficients belong to a field of characteristic other than 2 or 3, but must be modified for characteristic 2 or 3, because of the involved divisions by 2 and 3.
The reduction to a depressed cubic works for characteristic 2, but not for characteristic 3. However, in both cases, it is simpler to establish and state the results for the general cubic. The main tool for that is the fact that a multiple root is a common root of the polynomial and its formal derivative. In these characteristics, if the derivative is not a constant, it is a linear polynomial in characteristic 3, and is the square of a linear polynomial in characteristic 2. Therefore, for either characteristic 2 or 3, the derivative has only one root. This allows computing the multiple root, and the third root can be deduced from the sum of the roots, which is provided by Vieta's formulas.
A difference with other characteristics is that, in characteristic 2, the formula for a double root involves a square root, and, in characteristic 3, the formula for a triple root involves a cube root.
Cardano's formula
Cardano's result is that if
is a cubic equation such that p and q are real numbers such that is positive (this implies that the discriminant of the equation is negative) then the equation has the real root
where and are the two numbers and
See § Derivation of the roots, below, for several methods for getting this result.
As shown in
If there are three real roots, but Galois theory allows proving that, if there is no rational root, the roots cannot be expressed by an algebraic expression involving only real numbers. Therefore, the equation cannot be solved in this case with the knowledge of Cardano's time. This case has thus been called casus irreducibilis, meaning irreducible case in Latin.
In casus irreducibilis, Cardano's formula can still be used, but some care is needed in the use of cube roots. A first method is to define the symbols and as representing the principal values of the root function (that is the root that has the largest real part). With this convention Cardano's formula for the three roots remains valid, but is not purely algebraic, as the definition of a principal part is not purely algebraic, since it involves inequalities for comparing real parts. Also, the use of principal cube root may give a wrong result if the coefficients are non-real complex numbers. Moreover, if the coefficients belong to another field, the principal cube root is not defined in general.
The second way for making Cardano's formula always correct, is to remark that the product of the two cube roots must be –p / 3. It results that a root of the equation is
In this formula, the symbols and denote any square root and any cube root. The other roots of the equation are obtained either by changing of cube root or, equivalently, by multiplying the cube root by a primitive cube root of unity, that is
This formula for the roots is always correct except when p = q = 0, with the proviso that if p = 0, the square root is chosen so that C ≠ 0. However, Cardano's formula is useless if as the roots are the cube roots of Similarly, the formula is also useless in the cases where no cube root is needed, that is when the cubic polynomial is not irreducible; this includes the case
This formula is also correct when p and q belong to any field of characteristic other than 2 or 3.
General cubic formula
A cubic formula for the roots of the general cubic equation (with a ≠ 0)
can be deduced from every variant of Cardano's formula by reduction to a depressed cubic. The variant that is presented here is valid not only for real coefficients, but also for coefficients a, b, c, d belonging to any field of characteristic other than 2 or 3.
The formula being rather complicated, it is worth splitting it in smaller formulas.
Let
(Both and can be expressed as resultants of the cubic and its derivatives: is −1/8a times the resultant of the cubic and its second derivative, and is −1/12a times the resultant of the first and second derivatives of the cubic polynomial.)
Then let
where the symbols and are interpreted as any square root and any cube root, respectively (every nonzero complex number has two square roots and three cubic roots). The sign "±" before the square root is either "+" or "–"; the choice is almost arbitrary, and changing it amounts to choosing a different square root. However, if a choice yields C = 0 (this occurs if ), then the other sign must be selected instead. If both choices yield C = 0, that is, if a fraction 0/0 occurs in following formulas; this fraction must be interpreted as equal to zero (see the end of this section). With these conventions, one of the roots is
The other two roots can be obtained by changing the choice of the cube root in the definition of C, or, equivalently by multiplying C by a
where ξ = –1 + √–3/2.
As for the special case of a depressed cubic, this formula applies but is useless when the roots can be expressed without cube roots. In particular, if the formula gives that the three roots equal which means that the cubic polynomial can be factored as A straightforward computation allows verifying that the existence of this factorization is equivalent with
Trigonometric and hyperbolic solutions
Trigonometric solution for three real roots
When a cubic equation with real coefficients has three real roots, the formulas expressing these roots in terms of radicals involve complex numbers.
are[26]
This formula is due to François Viète.[22] It is purely real when the equation has three real roots (that is ). Otherwise, it is still correct but involves complex cosines and arccosines when there is only one real root, and it is nonsensical (division by zero) when p = 0.
This formula can be straightforwardly transformed into a formula for the roots of a general cubic equation, using the back-substitution described in § Depressed cubic.
The formula can be proved as follows: Starting from the equation t3 + pt + q = 0, let us set t = u cos θ. The idea is to choose u to make the equation coincide with the identity
For this, choose and divide the equation by This gives
Combining with the above identity, one gets
and the roots are thus
Hyperbolic solution for one real root
When there is only one real root (and p ≠ 0), this root can be similarly represented using
If p ≠ 0 and the inequalities on the right are not satisfied (the case of three real roots), the formulas remain valid but involve complex quantities.
When p = ±3, the above values of t0 are sometimes called the Chebyshev cube root.[29] More precisely, the values involving cosines and hyperbolic cosines define, when p = −3, the same analytic function denoted C1/3(q), which is the proper Chebyshev cube root. The value involving hyperbolic sines is similarly denoted S1/3(q), when p = 3.
Geometric solutions
Omar Khayyám's solution
For solving the cubic equation x3 + m2x = n where n > 0,
A simple modern proof is as follows. Multiplying the equation by x/m2 and regrouping the terms gives
The left-hand side is the value of y2 on the parabola. The equation of the circle being y2 + x(x − n/m2) = 0, the right hand side is the value of y2 on the circle.
Solution with angle trisector
A cubic equation with real coefficients can be solved geometrically using
A cubic equation can be solved by compass-and-straightedge construction (without trisector) if and only if it has a rational root. This implies that the old problems of angle trisection and doubling the cube, set by ancient Greek mathematicians, cannot be solved by compass-and-straightedge construction.
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.[22][31] When the cubic is written in depressed form (2), t3 + pt + q = 0, as shown above, the solution can be expressed as
Here is an angle in the unit circle; taking 1/3 of that angle corresponds to taking a cube root of a complex number; adding −k2π/3 for k = 1, 2 finds the other cube roots; and multiplying the cosines of these resulting angles by corrects for scale.
For the non-depressed case (1) (shown in the accompanying graph), the depressed case as indicated previously is obtained by defining t such that x = t − b/3a so t = x + b/3a. Graphically this corresponds to simply shifting the graph horizontally when changing between the variables t and x, without changing the angle relationships. This shift moves the point of inflection and the centre of the circle onto the y-axis. Consequently, the roots of the equation in t sum to zero.
One real root
In the Cartesian plane
When the graph of a
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 π/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 π/3, the major axis is vertical and its foci, the roots of the derivative, are complex conjugates. And if that angle is π/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.
Galois group
Given a cubic
). As these automorphisms must permute the roots of the polynomials, this group is either the group S3 of all six permutations of the three roots, or the group A3 of the three circular permutations.The discriminant Δ of the cubic is the square of
where a is the leading coefficient of the cubic, and r1, r2 and r3 are the three roots of the cubic. As changes of sign if two roots are exchanged, is fixed by the Galois group only if the Galois group is A3. In other words, the Galois group is A3 if and only if the discriminant is the square of an element of K.
As most integers are not squares, when working over the field Q of the rational numbers, the Galois group of most irreducible cubic polynomials is the group S3 with six elements. An example of a Galois group A3 with three elements is given by p(x) = x3 − 3x − 1, whose discriminant is 81 = 92.
Derivation of the roots
This section regroups several methods for deriving Cardano's formula.
Cardano's method
This method is due to
This method applies to a depressed cubic t3 + pt + q = 0. The idea is to introduce two variables u and such that and to substitute this in the depressed cubic, giving
At this point Cardano imposed the condition This removes the third term in previous equality, leading to the system of equations
Knowing the sum and the product of u3 and one deduces that they are the two solutions of the quadratic equation
so
The discriminant of this equation is , and assuming it is positive, real solutions to this equation are (after folding division by 4 under the square root):
So (without loss of generality in choosing u or ):
As the sum of the cube roots of these solutions is a root of the equation. That is
is a root of the equation; this is Cardano's formula.
This works well when but, if the square root appearing in the formula is not real. As a complex number has three cube roots, using Cardano's formula without care would provide nine roots, while a cubic equation cannot have more than three roots. This was clarified first by Rafael Bombelli in his book L'Algebra (1572). The solution is to use the fact that that is, This means that only one cube root needs to be computed, and leads to the second formula given in § Cardano's formula.
The other roots of the equation can be obtained by changing of cube root, or, equivalently, by multiplying the cube root by each of the two
Vieta's substitution
Vieta's substitution is a method introduced by François Viète (Vieta is his Latin name) in a text published posthumously in 1615, which provides directly the second formula of § Cardano's method, and avoids the problem of computing two different cube roots.[35]
Starting from the depressed cubic t3 + pt + q = 0, Vieta's substitution is t = w – p/3w.[b]
The substitution t = w – p/3w transforms the depressed cubic into
Multiplying by w3, one gets a quadratic equation in w3:
Let
be any nonzero root of this quadratic equation. If w1, w2 and w3 are the three cube roots of W, then the roots of the original depressed cubic are w1 − p/3w1, w2 − p/3w2, and w3 − p/3w3. The other root of the quadratic equation is This implies that changing the sign of the square root exchanges wi and − p/3wi for i = 1, 2, 3, and therefore does not change the roots. This method only fails when both roots of the quadratic equation are zero, that is when p = q = 0, in which case the only root of the depressed cubic is 0.
Lagrange's method
In his paper Réflexions sur la résolution algébrique des équations ("Thoughts on the algebraic solving of equations"),
In the case of cubic equations, Lagrange's method gives the same solution as Cardano's. Lagrange's method can be applied directly to the general cubic equation ax3 + bx2 + cx + d = 0, but the computation is simpler with the depressed cubic equation, t3 + pt + q = 0.
Lagrange's main idea was to work with the
be the discrete Fourier transform of the roots. If s0, s1 and s2 are known, the roots may be recovered from them with the inverse Fourier transform consisting of inverting this linear transformation; that is,
By Vieta's formulas, s0 is known to be zero in the case of a depressed cubic, and −b/a for the general cubic. So, only s1 and s2 need to be computed. They are not symmetric functions of the roots (exchanging x1 and x2 exchanges also s1 and s2), but some simple symmetric functions of s1 and s2 are also symmetric in the roots of the cubic equation to be solved. Thus these symmetric functions can be expressed in terms of the (known) coefficients of the original cubic, and this allows eventually expressing the si as roots of a polynomial with known coefficients. This works well for every degree, but, in degrees higher than four, the resulting polynomial that has the si as roots has a degree higher than that of the initial polynomial, and is therefore unhelpful for solving. This is the reason for which Lagrange's method fails in degrees five and higher.
In the case of a cubic equation, and are such symmetric polynomials (see below). It follows that and are the two roots of the quadratic equation Thus the resolution of the equation may be finished exactly as with Cardano's method, with and in place of u and
In the case of the depressed cubic, one has and while in Cardano's method we have set and Thus, up to the exchange of u and we have and In other words, in this case, Cardano's method 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.
Computation of S and P
A straightforward computation using the relations ξ3 = 1 and ξ2 + ξ + 1 = 0 gives
This shows that P and S are symmetric functions of the roots. Using Newton's identities, it is straightforward to express them in terms of the elementary symmetric functions of the roots, giving
with e1 = 0, e2 = p and e3 = −q in the case of a depressed cubic, and e1 = −b/a, e2 = c/a and e3 = −d/a, in the general case.
Applications
Cubic equations arise in various other contexts.
In mathematics
- Angle trisection and doubling the cube are two ancient problems of geometry that have been proved to not be solvable by straightedge and compass construction, because they are equivalent to solving a cubic equation.
- first derivativeof this cubic are the complex coordinates of those foci.
- The circumradius of a heptagonal triangleis one of the solutions of a cubic equation. The values of trigonometric functions of angles related to satisfy cubic equations.
- Given the cosine (or other trigonometric function) of an arbitrary angle, the cosine of one-third of that angle is one of the roots of a cubic.
- The solution of the general quartic equation relies on the solution of its resolvent cubic.
- The eigenvalues of a 3×3 matrix are the roots of a cubic polynomial which is the characteristic polynomialof the matrix.
- The difference equationis a cubic equation.
- Intersection points of cubic Bézier curve and straight line can be computed using direct cubic equation representing Bézier curve.
- Critical points of a quartic function are found by solving a cubic equation (the derivative set equal to zero).
- Inflection points of a quintic function are the solution of a cubic equation (the second derivative set equal to zero).
In other sciences
- In analytical chemistry, the Charlot equation, which can be used to find the pH of buffer solutions, can be solved using a cubic equation.
- In thermodynamics, equations of state (which relate pressure, volume, and temperature of a substances), e.g. the Van der Waals equation of state, are cubic in the volume.
- Kinematic equations involving linear rates of acceleration are cubic.
- The speed of seismic Rayleigh waves is a solution of the Rayleigh wave cubic equation.
- The steady state speed of a vehicle moving on a slope with air friction for a given input power is solved by a depressed cubic equation.
- Kepler's third law of planetary motion is cubic in the semi-major axis.
Notes
- trigonometric tables. Textually: If the seeker is satisfied with an estimate, it is up to him to look into the table of chords of Almagest, or the table of sines and versed sines of Mothmed Observatory. This is followed by a short description of this alternate method (seven lines).
- ^ More precisely, Vieta introduced a new variable w and imposed the condition w(t + w) = p/3. This is equivalent with the substitution t = p/3w – w, and differs from the substitution that is used here only by a change of sign of w. This change of sign allows getting directly the formulas of § Cardano's formula.
References
- ISBN 978-3-0348-8599-7
- ^ ISBN 978-0-19-853936-0.
- ^ ISBN 0-387-12159-5
- ISBN 978-1-118-46029-0.
- ISBN 978-0-313-29497-6.
- ISBN 978-0-470-27797-3.
- ^ Guilbeau (1930, p. 8) states that "the Egyptians considered the solution impossible, but the Greeks came nearer to a solution."
- ^ a b Guilbeau (1930, pp. 8–9)
- ISBN 978-1578987542.
- ISBN 978-1603860512.
- ISBN 978-0-8284-0149-4
- ^ A paper of Omar Khayyam, Scripta Math. 26 (1963), pages 323–337
- MacTutor History of Mathematics archive, states, "Khayyam himself seems to have been the first to conceive a general theory of cubic equations."
- ^ Guilbeau (1930, p. 9) states, "Omar Al Hay of Chorassan, about 1079 AD did most to elevate to a method the solution of the algebraic equations by intersecting conics."
- ISBN 978-1-4939-3780-6.
- ISBN 81-86050-86-8
- ^ O'Connor, John J.; Robertson, Edmund F., "Sharaf al-Din al-Muzaffar al-Tusi", MacTutor History of Mathematics Archive, University of St Andrews
- JSTOR 604533
- ^ O'Connor, John J.; Robertson, Edmund F., "Fibonacci", MacTutor History of Mathematics Archive, University of St Andrews
- ISBN 9780321016188.
- S2CID 189888034
- ^ S2CID 124980170.
- ISBN 9781974130924.
...if two roots are imaginary, the product is positive...
- ^ "Solution for a depressed cubic equation". Math solution. Retrieved 2022-11-23.
- S2CID 125986006.
- ISBN 0-87819-622-6.
- ^ These are Formulas (80) and (83) of Weisstein, Eric W. 'Cubic Formula'. From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CubicFormula.html, rewritten for having a coherent notation.
- Mathematical Gazette86. November 2002, 473–477.
- ^ Abramowitz, Milton; Stegun, Irene A., eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover (1965), chap. 22 p. 773
- JSTOR 2323624. Archived from the original(PDF) on 2015-12-19.
- S2CID 172730765See esp. Fig. 2.
- JSTOR 2301359
- JSTOR 2972885
- JSTOR 1967772
- ISBN 3-540-13610-X
- Serret, Joseph-Alfred(ed.), Œuvres de Lagrange, vol. III, Gauthier-Villars, pp. 205–421
- ^ ISBN 3-540-43826-2
References
- Guilbeau, Lucye (1930), "The History of the Solution of the Cubic Equation", Mathematics News Letter, 5 (4): 8–12, JSTOR 3027812
Further reading
- Anglin, W. S.; Lambek, Joachim (1995), "Mathematics in the Renaissance", The Heritage of Thales, Springers, pp. 125–131, ISBN 978-0-387-94544-6Ch. 24.
- Dence, T. (November 1997), "Cubics, chaos and Newton's method", Mathematical Gazette, 81 (492), S2CID 125196796
- Dunnett, R. (November 1994), "Newton–Raphson and the cubic", Mathematical Gazette, 78 (483), S2CID 125643035
- ISBN 978-0-486-47189-1
- Mitchell, D. W. (November 2007), "Solving cubics by solving triangles", Mathematical Gazette, 91, S2CID 124710259
- Mitchell, D. W. (November 2009), "Powers of φ as roots of cubics", Mathematical Gazette, 93, S2CID 126286653
- Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007), "Section 5.6 Quadratic and Cubic Equations", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
- Rechtschaffen, Edgar (July 2008), "Real roots of cubics: Explicit formula for quasi-solutions", Mathematical Gazette, 92, S2CID 125870578
- Zucker, I. J. (July 2008), "The cubic equation – a new look at the irreducible case", Mathematical Gazette, 92, S2CID 125986006
External links
- "Cardano formula", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- History of quadratic, cubic and quartic equations on MacTutor archive.
- 500 years of NOT teaching THE CUBIC FORMULA. What is it they think you can't handle? – quartic equations