an( l infinite Series . 
^-111=11^-!, s = W 7 -c = ^TTTTTTJ s 
*+v'-l,andrf=J/fr 7 = ^^T 777 n = 2 _ v _ t 
Wherefore r = r + </= 4 the firft root; and 
— liu? — 5 - d / / , 
2 ^ ~ 1~ V ~2 2=tv / -i.v/-3 : _ 2± ^ 
the other two roots, which are alfo real. 
6o.Ex. S .The equations - 6* = 4 gi ve s * = _ a , and 
*= 2 ; therefor e c=^+ ^ = v^Tg = s/- 4 = 2 v/- r , 
J-v ^ + c = j^2 + 2v/-i = - i +v/-i,and d=V~ c 
~^ 2 ~ 2v/ ~ i = - i - v/- r. And hence 
r - s + d- -2 the firft root; and 
*— 5 + ^ - 4 - J ^ / __ / 
a ~ ~J~ ~3 - i ± v - i . V - 3 = r ^ v/3 which 
are the two extremes, or the greateft and leaft roots. So 
that in this example, card an’s rule gives the middle root. 
6 1 . Ex. 6. Let the equation be a 3 -gx = - 10. Then 
a ~ ~3 and b - - 5 ; fo that c = _ 2 y 
~ v/ ~ 2 ’ J = v? / ^ + c = ^~ 5 + ^-2 = 1 + v/-2, and 
d - Vb - c - ^-5 - V— 2 = 1 _ v / _2. 
Hence r = s + d= 2 tlie middle root; and 
2 ~ -3 - 1 - v '- 2 . \/-3 a-i i -v/6 
the greateft and leaft roots. 
62. Ex. 7. Take the equation # 3 - iaar = 9. Here 
a “ ~ 4 > a nd £ = f ; therefore <? = + # 3 = \/iT^ ^4 
I i i 2 - 
