83

ON CUBIC EQUATIONS

TRACT 28 -

where the two denominators under the

n±3 v urpi’

radicals differ by 4, we can assign the value of n, the aboveformula will give us the roots.

38. As, if the equation be x 1, — \8x = — 27. Here p — 18,

and q = 27 •, then = s/ V - = a/ 9 = 3, and V— = /27 =S also; therefore n + 3 = 8, or n — 1 = 4, either of whichgives n = 5: consequently, r = | ^ ={% = 3,

is the middle root, because ~ is found between the 4th and

’ 4 p

12th cases, which are the limits of the middle roots: and— H’(l ± ^n) = — 3- (1 ± v/5) = 4-854102 and 1-854102,are the greatest and least roots. Or, these two roots maybe also found in the same manner from the table of forms,which contains all the roots of every equation, thus: by a

few trials we find \/~r: = nearly, and therefore

v '20 9o 16 Do J

20‘ 95c

—-r- = 1-854 is the least root, because here n = 17-95,

lt>'93p 3 7

which lies far above the limit for the least roots, which is atthe 4th form, where n is = 9. And lastly, v / —

¥

3-0557"

: Z;

¥

9443

nearly, and therefore = 4-854 is the greatest root,

because 3 ^ is found between the 12th and 13th forms.

* 94 - 43 ^) *

which are the limits between which lies the greatest root, ofevery equation that has all its roots real.

39. Again, let the equation be .r 3 + 2x =12. Here p = 2,

and q = 12 ; hence = \/2p = v '4 = 2, and J/j = /fy

= v/8 = 2 also; therefore n— 3 = 2, or n + I = 6, ei-ther of which gives n = 5. Consequently r = ^

= — = ~~ = 2, and the other two roots are — k r (1 ± V —n)

= - 1 (1 ± Z-5) = - 1 4 Z-5-

40. But it is only by trials that we find out a propervalue for n in such cases as these; and this is-perhaps at-

*