EIWATIONS, cubic solution of, by Car dan's rule. Let the equation be reduced to the form x3-9 x±r=0, where q and r may be positive or aegative.
Assume x = a ± b, then the equation becomes a —q X a +6 r=0, Or a3+b3-1-3a b Xa+b—qxa+b+ r = 0 ; and since we have two unknown quantities, a and 5, and have made only one supposition respecting them, viz. that a b = x, we are at liberty to make another ; let Sab—q=. 0, then the equation becomes a3-1-b3 ± r = 0 ; also, since 3 a b — q = 0, b3 a and by sub stitution, 03 + r = 0, or r a3 + 4 = 0, an equation of a dratic form ; and by completing the square, a‘ + r a3 and a3 +i=± therefore a3 = —q-2- and a = 2 4 27' 2-2- Al since 3 2 Also, since a + b3 r = 0, b3 =— and b=47-11± Jr' q 3 • there 2 27 fore 4 27 4 — 27 • We may observe that when the sign of t_ co • • 4in one part of the expression, rs positive, it is negative in the other, that x 2 4 27 r 2 Since b = , the value of x is also 2 4 2 r q 3 4 Ex. Let x3 ± 6 x — 20 = 0; here —6, r -= —20,x 10 + ? 108 + 10 — 108= 2.732—.732 =2.
Cor. 1. Having obtained one value of x, the equation may be depressed to a quadratic, and the other roots found.
Cor. 2. The possible values of a and h being discovered, the other roots are known without the solution of a quadra tic.
The values of the cube roots of a3 are a — — 3 a, and — 1 — — 3 a;and the values of the cube root of b3 are b, —1 +3 -- 3 b.
Hence it appears, that there are nine values of a + b, three only of which can answer the conditions of the equation, the others having been introduced by involu tion. These nine values are, 1. a -1- b. —1 — 3 2. a + 2 — 1 V —3 4. 3. a + 2 + b.
2 5. a+. b.
2 2— 1 +?— 3 — 1 — 6. a+ oi 2 27. —1 — 8. — 1 — .‘/ — + 2 29. — 1 3 2 In the operation we assume 3 a b that is, the product of the corresponding values of a and b is supposed to be possi. ble. This consideration excludes the 2d. 3d. 4th. 5th. 7th. and 9th. values of a + b, or 5 ; therefore the three roots of the equa. tion are — 1 + a — 1 — a+6 ±2 — 1 — 3b.
Cor. 3. This solution only extends to those cases in which the cubic has two impossible roots.
For if the roots be rn ± 3 n, and — 2. m, then — g (the sum of the products of every two with their signs changed) 3 nt' — 3 n, and = m' + n ; also, r (the product of all the roots with their signs changed) = 2 m3 — 6 m n, and = m3— 3 m n ; and by,invo r' = — 6 to n 9 n' 93 3m4 si+3m'n' +n' 27 .2 .-27— 4 — n X and — = — n X 3m'—n, a quantity manifest ly impossible, unless n be negative, that is, unless two roots of the proposed cubic be impossible.