BIQUADRATIC EQUATIONS ALGEBRA an imaginary form. Take, for instance, the equation y 3 - Gy + 4 = 0, of which the roots are found to be y= 4/2 + 2 y/l + 4/2-2 JT It will be found by actual involution that the imaginary expressions 2 + 2^-1 and 2-2^-1 are the cubes of - 1 + y - 1 and - 1 - y - 1 respectively, whence by substitution we find y = 2, y=l+ V3, andy=l- ^3". 93. We shall now prove, that as often as the roots of the equation x 3 + qx + r = are real, q is negative, and -~^q 3 greater than ~r 2 ; and, conversely, that if ~q 3 be greater than ) s , the roots are all real. Let us suppose a to be a real root of the proposed equation Then x 3 + qx + r = , And a 3 + qa + r = . And therefore, by subtraction, x 3 - a? + q(x - a) = ; hence, dividing by x a, we have x 2 + ax + a" 2 + q . This quadratic equation is formed from the two remain ing roots of the proposed equation, and by resolving it we find _ _ x= -|a *J~{u~-q. And as, by hypothesis, all the roots are real, it is evident that q must necessarily be negative, and greater than {a- ; for otherwise the expression J - fa 2 - q would be imaginary. Let us change the sign of q, and put q a- + d ; thus the roots of the equation x 3 - qx + r = will be i -i+ /< , -f- M, and here a" is a positive quantity. To find an expression for r in terms of a and d, let fa 2 + d be substituted for q in the equation a 3 - </a + r = 0; we thence find r = - {a 3 + <: : so that to compare together the quantities q and r, we have these equations, be In order to make this comparison, let the cube of ~ taken, also the square of ~r, the results are and therefore, by subtraction, Now the square of any real quantity being always positive, it follows that 3d(a- - d)~ will be positive when d is posi tive; hence it is evident that in this case -q 3 must be greater than r~, and that ~g 3 cannot be less than ^r 2 , un less d be negative, that is, -unless - a + s /c/, - {a - V A/, the two other roots of the equation are imaginary. If we suppose d = 0, then -^q 3 = ~r-; and the roots of the equa tions, which in this case are also real, two of them being equal. Upon the whole, therefore, we infer, that since a cubic equation has always one real root, its roots will be all real as often as q is negative, and ^q 3 greater than r 2 ; and consequently, that in this case the formulas for the roots must express real quantities, notwithstanding their imagin ary form. 94. Let y 3 - qy + r = denote any equation of the form which has been considered in last article, namely, that which has its roots all real; then, if we put a= r, i2 = ___U2 J - i;- 2 , one of the roots, as expressed by the first formula (Art. 90) will be y= 4/a + b V ^T + Z/a-bJ^I. This expression, although under an imaginary form, must (as we have shown in last article) represent a real quantity, although we cannot obtain it by the ordinary process of arithmetic. The case of cubic equations, in which the roots are all real, is now called the irreducible case. It is remarkable that the expression - b / - 1 b - 1 and in general, _ I/a + b ,/^l + tja where n is any power of 2, admits of being reduced to another form, in which no impossible quantity is found. Thus, N /o~+ b and 4 + a - b y~l = x /2 a + 2 Ja 2 + b 2 , as is easily proved by first squaring and then taking the square root of the imaginary formulae. But when n is 3, it does not seem that such reduction can possibly take place. If each of the surds be expanded into an infinite series, and their sum be taken, the imaginary quantity J - 1 will vanish, and thus the root may be found by a direct process. SECT. XII. SOLUTION OF BIQUADRATIC EQUATIONS. 95. When a biquadratic equation contains all its terms, it has this form, where A, B, C, D denote any known quantities whatever. We shall first consider pure biquadratics, or such as con tain only the first and last terms, and therefore are of this form, x* = b*. In this case it is evident that x may be readily had by two extractions of the square root ; by the first we find x 2 b 2 , and by the second x = b. This, how ever, is only one of the values which x may have; for since x* = b*, therefore x l -b* = Q; but x* - b* may be resolved into two factors x 2 - b 2 and x 2 + b 2 , each of which admits of a similar resolution ; for x 2 - b 2 = (x - b)(x + b) and x 2 + b 2 = (x-b J -I) (x + b J -1). Hence it appears that the equation x* - 6 4 = may also be expressed thus so that x may have these four values, + b, -b, +b J~l, - b N /~l , two of which are real, and the others imaginary. 96. Next to pure biquadratic equations, in respect of easiness of resolution, are such as want the second and fourth terms, and therefore have this form, x 4 + qx- + 5 = 0. These may be resolved in the manner of quadratic equa tions; for if we put y = x 2 , we have y- + <iy + s = , from which we find y - , and therefore - q = 97. When a biquadratic equation has all its terms, tho manner of resolving it is not so obvious as in the two for mer cases, but its resolution may be always reduced to that
of a cubic equation. There are various methods by which
