3 Therefore, x=(+q+√q2±p3)+(q-√qp3) (C) (D) Or 1 x=(+q/q2±p3)3+ (±q±√q2±p3) These formulas are familiarly known among mathematicians as Carden's rule. (Art. 160.) When p is negative, and its cube greater than q2 the expression √ q2-p3 becomes imaginary; but we must not conclude, as some have, that the value of x is therefore imaginary, for admitting the expression q2-p3 imaginary, it can be represented by a-1, and the value of x in equation (C) will be 3 Now by actually extracting the root of these binomials, and adding the results together, the terms containing √-1 will destroy each other, and their sum will be a real quantity, and of course the value of x will become real; but the practical operation of expanding may be tedious and difficult. If necessary in any particular case, in order to make the series converge, change the terms of the binomial, and make stand first, and 1 second. q EXAMPLES. Given x2-6x=5.6, to find the value of x. Here, 3p6, and 2q=5.6, or p=-2, and q=2.8. Then x=(2.8+√7.84-8)+(2.8-7.84-8), by equation (C) Or a=(2.8+.4-1)+(2.8.4√1) Or *(2.8)=(1+√-1)+(1-1√−1) Expand the binomials by the binomial theorem, Art. (135,) and for sake of brevity, represent +/-1 by b; Then b2=-, and 64=1× Therefore, 3/2.8 3.6 3.6.9 3.6.9.12 3.6.9.12 =2+0.004535-0.000034=2,0045 1 =2.0045 or x=(2.0045) (2.8)=2.8256, nearly. (Art. 161.) Every cubic equation of the form of x3±px=±q has three roots, and their algebraic sum is 0. If the roots be represented by a, b and c, we shall have a+b+c=0, and abc=±q. If any two of these roots be equal, as b=c, then a=-2b (1), and ab2=±q (2). Putting the value of a taken from equation (1), into equation (2), and we have -2b3=±3/q. Hence, in case of there being two equal roots, such roots must each equal the cube root of one half the quantity represented by q. EXAMPLES. The equation x3-48x=128 has two equal roots; what are the roots? Here, -263-128, or b3=-64; therefore, b=-4. Two of the roots are each equal to -4, and as the sum of the three roots must be 0, therefore 448 must be the three roots. If the equation x3-27x= 54 have two equal roots, what are the roots? Ans. -3 -3 and +6. Either of these roots can be taken to verify the equation, and if they do not verify it, the equation has not equal roots. (Art. 162.) If a cubic equation in the form of x3+px2+qx+r=0 have two equal roots, each one of the equal roots will be equal to (p/p2-3q). The other root will be twice this quantity subtracted from p, because the sum of the three roots equal p, (Art. 157.) This expression is obtained from the consideration that the three roots represented by a, b and c, must form the following equations, (Art. 157.): abc=r (3) On the assumption that two of these roots are equal, that is, a=b, equations (1) and (2) become And a2+2ac=q (5) Multiply equation (4) by 2a, and we have Subtract (5) a2+2ac= q (8) This equation is a quadratic in relation to the root a, and a solution gives a=1(p±√p2-3q). px=±q (Art. 163.) A cubic equation in the form of can be resolved as a quadratic, in all cases in which q can be resolved into two factors, m and n, of such a magnitude that m2+p=n. For the values of pand q in the general equation, put the assumed values, mn=q, and p=n-m2. Then we have x2+nx-m2x=mn. Transpose -max, and then multiply both members by x, and x2+nx2=m2x2+mnx Add to both members, and extract square root; n2 4 n n Then x2+2=mx+이 Drop and divide by æ, and x=m. Therefore, if such factors of q can be found, the equation is already resolved, as a will be equal to the fac tor m. EXAMPLES. 1. Given x3+6x=88, to find the values of x. Here, mn=88=4×22 42+6=22. Hence, x=4. 2. Given r3-3x=14, to find one value of x. 2×7=14, 22+3=7. Hence, 2-2. 3. Given x3+6x=45, to find one value of x. Ans. x=3. (Art. 164.) A vast amount of labor has been expended by almost every mathematician, in striving to find direct solutions to the higher equations; but their efforts have not been successful beyond the fourth degree, and even in that degree the indirect methods of solution are in practice far less troublesome and tedious than the direct. We shall therefore consider some general properties of equations which belong to every degree. All the higher equations may be conceived to have been formed by the multiplication of the unknown quantity connected to each of the roots of the equation with a contrary sign, as shown in (Art. 157.) Let a, b, c, d, e, &c., be roots of an equation, and æ its unknown quantity, then the equation may be formed by the product of (x-a) (x-b) (x-c), &c., which product we may represent by Now it being admitted that equations can be thus formed by the multiplication of the unknown quantity joined to its roots, conversely, when any of its roots can be found, such root, with its contrary sign joined to the unknown term, will form a complete divisor for the equation, and by the |