144 ^ Treatise of Algebra.

x b \

whence s (=.<*+ \p z + — J and q (= ; 4- —

26

—) are also known. And by extracting the Roots os

2 P

the two assumed quadratic Equations * 2 + p* + ? = a

and * 2 -{- rx + s = o, we have x equal to-+

/p z r /r z

*/ - q, in the one, and equal to - — + - s.

4 2 4

P /P l

in the other, winch last is equal to — + v'-r, be-

- 4

cause r=-/>. Therefore the four Roots of the Bi-

causer=-/>. Therefore the four Roots of the Bi-

quadratic, jt + » -f*<7x 2 -}-Ax-|-c=o, are

example.

Let the Equation propounded be y* - 4y 3 - 8y -f- 3 2— 0 - then, first, to take away the second Term there-of, let at —J— 1 be substituted for y, and it will become* — 6 x 1 — 16* 4- 21 = o ; which being comparedwith the general Equation, x 4 * + ax 1 -f- bx + c = o,we have a = — 6, b=z — 16 and c = 21, and conse-

quently p 6 — lip* — 48 p 1 (r= p 6 4* 2apl

25 6 (= I 1 ). Now, in order, also, to destroy the se-cond Term of the last Equation, make z -f- 4. = p z ,and then, this Value being substituted above, you willhave z 3 — 962 — 576 : Whence, by the Method above

explained, z will Hp sound (=2 28S +V^8]a-pl3

1