1. When a biquadratic equation contains all its terms, it has this
form,
x^4 + Ax^3 + Bx^2 + Cx + 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[root]-1)(x + b[root]-1). Hence it appears that the
equation x^4 - b^4 = 0 may also be expressed thus,
(x - b)(x + b)(x - b[root]-1)(x + b[root]-1) = 0;
so that x may have these four values,
+b, -b, +b[root]-1, -b[root]-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 = 1/2{-q [+-] [root](q^2 - 4s)}, and therefore
x = [+-][root]1/2{-q [+-] [root](q^2 - 4s)}.
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 =
[root]a + [root]b + [root]c, and also suppose that the letters a, b, c
denote the roots of the cubic equation
z^3 + Pz^2 + Qz - R = 0;
then, from the theory of equations we have
a + b + c = -P, ab + ac + bc = Q, abc = R.