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ON CUBIC EQUATIONS

TRACT 28 ,

^ a = l/(±q + v'Kt?) 2 + (|p) 3 ]) X 1 or x -(y = Ki! - y/i (my + (IP) 3 ]) x 1 or x -

or x —

I:+V-3

1+a/-3

the three values of a and y ; for every quantity has three,different forms of the cube root, and the cube root of 1, is

not only 1, but also — 3 or _ l— ^ 3 . Hence then the

J ’ 2 2

three values of a + y or x, or the three roots of the equa-tion x 3 + px — q, are

v'Us , + v / [()?)"+ (ip) 3 ]) x 1 or X or x -- -b

i + V'- 1 ’

- VlilqY + (Ip) 3 ]) x ior x 3

- or x —

ivhere the signs of a/ — 3 must be opposite, in the values

of z and y, that is, when it is - in the one, it must be

3 in the other, otherwise their product zy will not

be = — -j-p, as it ought to be.

42, Or if we put a — ^p, and b~\q , the same threeroots will be

ftb + y/{b z + a 3 )] -\-i/[b - y/{b'- + a 3 )] = the 1st root or r,-mb+^/^+a^K 1-^-3)

- \\/{b - V(b z + a 3 )]. (1 + */ - 3) the 2d root.

-!v'[6+v/('x + « 3 )]-( 1 +^“ 3)

— x\/\b — s/{b z -(- a 3 )]. (1 — v/ — 3) the 3d root.

43. Or again, the 1st root r being

l/\b + </{b z + a 3 )] + l/\b — */{& + a 3 )], the other two

will be

- i r + ^!/V> + V (4* + « 3 )] - “ v 7 (4* 4 - a 2 )] =

the 2d root, and

' - xr - ^- 3 t/[4 + y/V + « 3 )] + ~ y/(P+ a3 )'l =

the 3d root.

44. Or, if tve put s = l/\b + y/[b z + a3 )J> and d ==i ]/\b — */(b z -f a 3 )], the roots will be