# 1911 Encyclopædia Britannica/Equation/Biquadratic Equations

*Biquadratic Equations*.

1. When a biquadratic equation contains all its terms, it has this form,

*x*

^{4}+ A

*x*

^{3}+ B

*x*

^{2}+ C

*x*+ D = 0,

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, *x*^{4} = *b*^{4}. In
this case it is evident that *x* may be readily had by two extractions of
the square root; by the first we find *x*^{2} = *b*^{2}, and by the second *x* = b.
This, however, is only one of the values which *x* may have; for since
*x*^{4} = *b*^{4}, therefore *x*^{4} − *b*^{4} = 0; but *x*^{4} − *b*^{4} may be resolved into two
factors *x*^{2} − *b*^{2} and *x*^{2} + *b*^{2}, each of which admits of a similar resolution;
for *x*^{2} − *b*^{2} = (*x* − *b*)(*x* + *b*) and *x*^{2} + *b*^{2} = (*x* − *b*√−1)(*x* + *b*√−1).
Hence it appears that the equation *x*^{4} − *b*^{4} = 0 may also be expressed
thus,

*x*−

*b*) (

*x*+

*b*) (

*x*−

*b*√−1) (

*x*+

*b*√−1) = 0;

so that *x* may have these four values,

*b*, −

*b*, +

*b*√−1, −

*b*√−1,

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,

*x*

^{4}+

*qx*

^{2}+

*s*= 0.

These may be resolved in the manner of quadratic equations; for if
we put *y* = *x*^{2}, we have

*y*

^{2}+

*qy*+

*s*= 0,

from which we find *y* = 12 {−*q* ± √(*q*^{2} − 4*s*) }, and therefore

*x*= ±√12 {−

*q*± √(

*q*

^{2}− 4

*s*) }.

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,

*y*

^{4}+

*py*

^{2}+

*qy*+

*r*= 0,

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

*z*

^{3}+ P

*z*

^{2}+ Q

*z*− R = 0;

then, from the theory of equations we have

*a*+

*b*+

*c*= −P,

*ab*+

*ac*+

*bc*= Q,

*abc*= R.

We square the assumed formula

*y*= √

*a*+ √

*b*+ √

*c*,

and obtain *y*^{2} = *a* + *b* + *c* + 2(√*ab* + √*ac* + √*bc*);

or, substituting −P for *a* + *b* + *c*, and transposing,

*y*

^{2}+ P = 2(√

*ab*+ √

*ac*+ √

*bc*).

Let this equation be also squared, and we have

*y*

^{4}+ 2P

*y*

^{2}+ P

^{2}= 4 (

*ab*+

*ac*+

*bc*) + 8 (√

*a*

^{2}

*bc*+ √

*ab*

^{2}c + √

*abc*

^{2});

and since *ab* + *ac* + *bc* = Q,

and √*a*^{2}*bc* + √*ab*^{2}c + √*abc*^{2} = √*abc* (√*a* + √*b* + √*c*) = √R·*y*,

the same equation may be expressed thus:

*y*

^{4}+ 2P

*y*

^{2}+ P

^{2}= 4Q + 8√R·

*y*.

Thus we have the biquadratic equation

*y*

^{4}+ 2P

*y*

^{2}− 8√R·y + P

^{2}− 4Q = 0,

one of the roots of which is *y* = √*a* + √*b* + √*c*, while *a*, *b*, *c* are the
roots of the cubic equation *z*^{3} + P*z*^{2} + Q*z* − R = 0.

4. In order to apply this resolution to the proposed equation
*y*^{4} + *py*^{2} + *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

*y*

^{4}+

*py*

^{2}+

*qy*+

*r*= 0,

*y*

^{4}+ 2P

*y*

^{2}− 8√R

*y*+ P

^{2}− 4Q = 0,

and it immediately appears that

*p*, −8√R =

*q*, P

^{2}− 4Q =

*r*;

and from these equations we find

*p*, Q = 116 (

*p*

^{2}− 4

*r*), R = 164

*q*

^{2}.

Hence it follows that the roots of the proposed equation are generally expressed by the formula

*y*= √

*a*+ √

*b*+ √

*c*;

where *a*, *b*, *c* denote the roots of this cubic equation,

z^{3} + | p |
z^{2} + | p^{2} − 4r |
z − | q^{2} |
= 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

z^{3} + | p |
z^{2} + | p^{2} − 4r |
z − | q^{2} |
= 0 |

2 | 16 | 64 |

becomes, after reduction,

*v*

^{3}+ 2

*pv*

^{2}+ (

*p*

^{2}− 4

*r*)

*v*−

*q*

^{2}= 0;

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 *y*^{4} + *py*^{2} + *qy* + *r* = 0 be any biquadratic equation wanting its
second term. Form this cubic equation

*v*

^{3}+ 2

*pv*

^{2}+ (

*p*

^{2}− 4

*r*)

*v*−

*q*

^{2}= 0,

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.)