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,
x4 + qx² + s = 0.
These may be resolved in the manner of quadratic equations; for if we put y = x², we have
y² + qy + s = 0,
from which we find y = ½ {−q ± √(q² − 4s) }, and therefore
x = ±√½ {−q ± √(q² − 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,
y4 + py² + qy + r = 0,
where the second term is wanting, and where p, q, r denote any known quantities whatever.