A L G E B K A [CUBIC EQUATIONS. and y = v + z Thus we have obtained a value of the unknown quantity y, in terms of the known quantities <? and r; therefore the equation is resolved. 90. But this is only one of three values which y may have. Let us, for the sake of brevity, put A = -*r+ and put then, from what has been shown (Art. 88), it is evident that v and z have each these three values, To determine the corresponding values of v and z, we must consider that vz = - f = ^ AB~. Now if we observe O that a/3= 1, it will immediately appear that v + z has these three values, which are therefore the three values of y. The first of these formulae is commonly known by the name of Cardan s rule; but it is well known that Cardan was not the inventor, and that it ought to be attributed to Nicholas Tartalea and Scipio Ferreus, who discovered it much about the same time, and independently of each other. (See the Historical Introduction.) The formulae given above for the roots of a cubic equa tion may be put under a different form, better adapted to the purposes of arithmetical calculation, as follows : Because vz= - 1, therefore z= -| x - = - -&= ; hence v + z= --^r : thus it appears that the three values VA of y may also be expressed thus : 91. To show the manner of applying these formulae, let it be required to determine x from the cubic equation As this equation has all its terms, the first step towards its resolution is to transform it into another which shall want the second term, by substituting y - 1 for x as directed (Art. 84). The operation will stand thus : 3 + Qx = + 9y - 9 -13= -13 .-. adding these, the transformed equation is 2/3 - 20 = 0, which being compared with the general equation, 3/ 3 + qy + r = , gives i = G, r = - 20 ; hence therefore the second formula of last article gives y- - _ _ . 2 V10+ J108- 3 /._ ~7F^: ; but as this expression m- /V 10+ V 108 ,-olves a radical quantity, let the square root of 108 be
- aken and added to 10, and the cube root of the sum
- ound; thus
have therefore 2-732 nearlv > and = 732 ; hence we at last find ., _ V10 + V108 one of the values of y to be 2 "732 - "732 = 2 . In finding the cube root of the radical quantity ~ we liave takeu its a PP roximate value, so as to have the expression for the root under a rational form, and in this way we can always find, as near as we please, the cube root of any surd of the form a+ Jb, where 6 is a positive number. But it will sometimes happen that the cube root of such a surd can be expressed exactly by another surd of the same form; and accord ingly, in the present case, it appears that the cube root of10+ ^/TOSjs 1+ J3, as may be proved by actually raising 1 + N /3 to the third power. Hence we find 2(1 -V3) -- _ that y=l+ J3 + 1 - x/3 = 2, as before. The other two values of y will be had by substituting 1 + J3 and 1 - fi for /A and -l| in the second and V A. third formulae of last article, and restoring the values of a and /3. We thus have _ - 1 - V - 3 x (1 + -l+V-3 So that the three values of y are + 2, -1+V^ -1-V-9J and since x = y-l, the corresponding values of x are + 1, -2+^-9, -2-V" 9 - Thus it appears that one of the roots of the proposed equation is real, and the other two imaginary. The two imaginary roots might have been found other wise, by considering that since one root of the equation is 1, the equation must be divisible by x - 1 (Art. 78). Accordingly, the division being actually performed, and the quotient put = 0, we have the quadratic equation, which gives x = - 2 J^T, the same imaginary values as before. 92. In the application of the preceding formulas to the resolution of the equation f + gy + r = 0, it is necessary to find the square root of -^<f + {r 2 ; now, when that quantity is positive, as in the equation y z + Gy - 20 = 0, which was resolved in last article, no difficulty occurs, for its root may be found either exactly or to as great a degree of accuracy as we please. As, however, the coefficients q and r are independent ot each other, it is evident that q may be negative, and such that -i*? 3 is greater than i?- 2 . In this case, the expression _i_ r/ 3 + if.2 w jn be negative, and therefore its square root an imaginary quantity ; so that all the roots appear under an imaginary form. But we are certain (Art. 81) that every cubic equation must have at least one real root. The truth
ia that roots are frequently real, though they appear und^r
