The first of these formulae is commonly known by the name of
Cardan’s rule (see Algebra: History).
The formulae given above for the roots of a cubic equation may be put under a different form, better adapted to the purposes of arithmetical calculation, as follows:—Because vz = −13 q, therefore z = −13q × 1/v = −13 q / ∛A; hence v + z = ∛A − 13 q / ∛A: thus it appears that the three values of y may also be expressed thus:
y = α∛A − 13 qβ / ∛A
See below, Theory of Equations, §§ 16 et seq.
1. When a biquadratic equation contains all its terms, it has this form,
where A, B, C, D denote known quantities.
We shall first consider pure biquadratics, or such as contain only the first and last terms, and therefore are of this form, x4 = b4. In this case it is evident that x may be readily had by two extractions of the square root; by the first we find x2 = b2, and by the second x = b. This, however, is only one of the values which x may have; for since x4 = b4, therefore x4 − b4 = 0; but x4 − b4 may be resolved into two factors x2 − b2 and x2 + b2, each of which admits of a similar resolution; for x2 − b2 = (x − b)(x + b) and x2 + b2 = (x − b√−1)(x + b√−1). Hence it appears that the equation x4 − b4 = 0 may also be expressed thus,
so that x may have these four values,
two of which are real, and the others imaginary.
2. Next to pure biquadratic equations, in respect of easiness of resolution, are such as want the second and fourth terms, and therefore have this form,
These may be resolved in the manner of quadratic equations; for if we put y = x2, we have
from which we find y = 12 {−q ± √(q2 − 4s) }, and therefore
3. When a biquadratic equation has all its terms, its resolution may be always reduced to that of a cubic equation. There are various methods by which such a reduction may be effected. The following was first given by Leonhard Euler in the Petersburg Commentaries, and afterwards explained more fully in his Elements of Algebra.
We have already explained how an equation which is complete in its terms may be transformed into another of the same degree, but which wants the second term; therefore any biquadratic equation may be reduced to this form,
where the second term is wanting, and where p, q, r denote any known quantities whatever.
That we may form an equation similar to the above, let us assume y = √a + √b + √c, and also suppose that the letters a, b, c denote the roots of the cubic equation
then, from the theory of equations we have
We square the assumed formula
and obtain y2 = a + b + c + 2(√ab + √ac + √bc);
or, substituting −P for a + b + c, and transposing,
Let this equation be also squared, and we have
and since ab + ac + bc = Q,
and √a2bc + √ab2c + √abc2 = √abc (√a + √b + √c) = √R·y,
the same equation may be expressed thus:
Thus we have the biquadratic equation
one of the roots of which is y = √a + √b + √c, while a, b, c are the roots of the cubic equation z3 + Pz2 + Qz − R = 0.
4. In order to apply this resolution to the proposed equation y4 + py2 + qy + r = 0, we must express the assumed coefficients P, Q, R by means of p, q, r, the coefficients of that equation. For this purpose let us compare the equations
y4 + 2Py2 − 8√Ry + P2 − 4Q = 0,
and it immediately appears that
and from these equations we find
Hence it follows that the roots of the proposed equation are generally expressed by the formula
where a, b, c denote the roots of this cubic equation,
| z3 + | p | z2 + | p2 − 4r | z − | q2 | = 0. |
| 2 | 16 | 64 |
But to find each particular root, we must consider, that as the square root of a number may be either positive or negative, so each of the quantities √a, √b, √c may have either the sign + or − prefixed to it; and hence our formula will give eight different expressions for the root. It is, however, to be observed, that as the product of the three quantities √a, √b, √c must be equal to √R or to −18 q; when q is positive, their product must be a negative quantity, and this can only be effected by making either one or three of them negative; again, when q is negative, their product must be a positive quantity; so that in this case they must either be all positive, or two of them must be negative. These considerations enable us to determine that four of the eight expressions for the root belong to the case in which q is positive, and the other four to that in which it is negative.
5. We shall now give the result of the preceding investigation in the form of a practical rule; and as the coefficients of the cubic equation which has been found involve fractions, we shall transform it into another, in which the coefficients are integers, by supposing z = 14 v. Thus the equation
| z3 + | p | z2 + | p2 − 4r | z − | q2 | = 0 |
| 2 | 16 | 64 |
becomes, after reduction,
it also follows, that if the roots of the latter equation are a, b, c, the roots of the former are 14 a, 14 b, 14 c, so that our rule may now be expressed thus:
Let y4 + py2 + qy + r = 0 be any biquadratic equation wanting its second term. Form this cubic equation
and find its roots, which let us denote by a, b, c.
Then the roots of the proposed biquadratic equation are,
| when q is negative, | when q is positive, |
| y = 12 (√a + √b + √c), | y = 12 (−√a − √b − √c), |
| y = 12 (√a − √b − √c), | y = 12 (−√a + √b + √c), |
| y = 12 (−√a + √b − √c), | y = 12 (√a − √b + √c), |
| y = 12 (−√a − √b + √c), | y = 12 (√a + √b − √c). |
See also below, Theory of Equations, § 17 et seq. (X.)
1. In the subject “Theory of Equations” the term equation is used to denote an equation of the form xn − p1xn−1 ... ± pn = 0, where p1, p2 ... pn are regarded as known, and x as a quantity to be determined; for shortness the equation is written ƒ(x) = 0.
The equation may be numerical; that is, the coefficients p1, p2n, ... pn are then numbers—understanding by number a quantity of the form α + βi (α and β having any positive or negative real values whatever, or say each of these is regarded as susceptible of continuous variation from an indefinitely large negative to an indefinitely large positive value), and i denoting √−1.
Or the equation may be algebraical; that is, the coefficients are not then restricted to denote, or are not explicitly considered as denoting, numbers.
1. We consider first numerical equations. (Real theory, 2-6; Imaginary theory, 7-10.)
2. Postponing all consideration of imaginaries, we take in the first instance the coefficients to be real, and attend only to the real roots (if any); that is, p1, p2, ... pn are real positive or negative quantities, and a root a, if it exists, is a positive or negative quantity such that an − p1an−1 ... ± pn = 0, or say, ƒ(a) = 0.
It is very useful to consider the curve y = ƒ(x),—or, what would come to the same, the curve Ay = ƒ(x),—but it is better to retain the first-mentioned form of equation, drawing, if need be, the ordinate y on a reduced scale. For instance, if the given equation be x3 − 6x2 + 11x − 6.06 = 0,[1] then the curve
- ↑ The coefficients were selected so that the roots might be nearly 1, 2, 3.
