116

ON CUBIC EQUATIONS.

ERACT 28}

Of the Roots bjj another Class of Series.

124 . But there are yet other series, converging muchfaster than those in the foregoing class, by the help ofwhich, and Cardan’s rule conjointly, may always be foundthe roots of those equations, in which that rule fails when itis applied singly, that is, in what is called the irreduciblecase, or that in which c" is negative. And these series arefound by introducing another cubic equation, having thesame values of b and <?, as the given equation, except that,in the new equation, the value of c 1 is positive, while in thegiven one it is negative. For when e A is positive, the newequation to which it belongs, has only one real root, andthat root is always found by Cardan’s rule; but the contrarytakes place when c 1 is negative, the equation having thenthree real roots, though they are not always determinableby that rule, because the radical quantities can seldom beextracted, on account of the square root of the negativequantity, which is contained in them.

125 . "Now the general expression for the root, by Car-dan’s rule, being s + d =%/(b + V ± c 1 ) + l/(b — >/ ± c l ), or

+ b) — v'l v'± £l ~ b)> if the cubic roots of eachof these be extracted, by the binomial theorem, as at Art.68, we shall obtain these four forms :

1. + + 1 - &c.

2. $/[b+f-c z ) +Z/{b- y'-c 2 )=2J/6 x : 1 + - &c.

3 . $/( v/ + + b )✓ +c'- b) = £ x : -1 + 5^ + &c.

4. l'(v'-c*+b)-l'(x/-c*-b)=^ X : - J

126 . Of which, the series in the first and third denotethe only real root of the equation, when c z is positive, ac-cording as c is greater or less than b, which root call x; andthe series in the second and fourth forms denote the greatestand least roots of the equation, when c 1 is negative, which.