See below, Theory of Equations, §§ 16 et seq.
IV. Biquadratic Equations.
1. When a biquadratic equation contains all its terms, it has this form,
x4 + Ax³ + Bx² + 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, 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,
(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.