| z³ + | p | z² + | p² − 4r | z − | q² | = 0 |
| 2 | 16 | 64 |
becomes, after reduction,
v³ + 2pv² + (p² − 4r) v − q² = 0;
it also follows, that if the roots of the latter equation are a, b, c, the roots of the former are ¼ a, ¼ b, ¼ 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
v³ + 2pv² + (p² − 4r) v − q² = 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 = ½ (√a + √b + √c), | y = ½ (−√a − √b − √c), |
| y = ½ (√a − √b − √c), | y = ½ (−√a + √b + √c), |
| y = ½ (−√a + √b − √c), | y = ½ (√a − √b + √c), |
| y = ½ (−√a − √b + √c), | y = ½ (√a + √b − √c). |
See also below, Theory of Equations, § 17 et seq.