A 'Treatift of Algebra. 143

the Solution of Problems, I shall take no further noticeof them here, but proceed to

‘The Solution of biquadratic Equations according toDes Carte* j Method.

First, let the second Term be destroyed (if need be)as in the Solution of Cubics; which being done, thegiven Equation will be reduced to this Form, viz. x** ax 1 -f- bx -f- c = o ; where a, b and c may repre-sent any Quantities whatever, po sitive or negative. As-sume •**+/>•*'•+- y X + r* + * = * -f- ax 1 + bx-\-c tor, which is the fame, let the proposed biquadratic Equa-tion be considered as produced by the Multiplication ofthe two Quadratics jr 2 -f - px + i — ° a °d x 2 -j- rx -J- s— o : Then, these last Expressions being actually mul-tiply'd into each other, we shall have *4 * -}- ax z bx

u; whence,

by equating the homologous Terrfts (in order to findthe Value of the assumed Coefficients- />,. q, r and s) weshall have p + r = o, s -f- q -j- pr = a, ps -f- qr—band qs — c j from the first of which we .have r —~p ;from the second, s -J- q ('= a — pr) =:«-(- p z j from theb

third, s — q— — j and from the fourth, qs—c\ let sour

• P

times the last of these be subtracted from the Square ofthe Precedent, and there will come out s 1 — zsq -f- q z

b

= a 2 2 ap z -}- p* — 4r; but s — q being — —, by the

P

third, therefore will j 2 — 2 sq q x be, also, equal tob z b z

— j consequently a z -f- tap 1 + p* — 4c = — and p° -+■

P z

tap*

> z zn b z -j from which p will be determin-

ed as in Example the second of the Solution of.Cubics;

whence