550 [BIQUADRATIC EQUATIONS. such a reduction may bo effected. The following, which we select as one of the most ingenious, was first given by Euler in the Petersburg Commentaries, and afterwards ex plained more fully in his Elements of Alyebra. We have already explained, Art. 92, how an equation which is complete in its terms may be transformed into another of the same degree, but which wants the second term; therefore any biquadratic equation may be reduced to this form, where the second term is wanting, and where p, q, r, de note any known quantities whatever. That we may form an equation similar to the above, let us assume y */a+ *Jb + *Jc, and also suppose that the letters a, b, c denote the roots of the cubic equation 2 3 + P2 2 + Q^ - R = ; then, from the theory of equations we have a + b + c = - P, ab + ac + bc = Q, abc = II . We square the assumed formula y= Va + /*+ >J C !_ and obtain y- = a + b + c + 2( ,Jab + v V<c + Jbc), or, substituting - P for a + b + c, and transposing; Let this equation be also squared, and we have y* + 2Py 2 + P 2 = 4(ab + ac + be) + 8(Vo 5 Fc + V and since ab + ac + be == Q, _ and V 2 6c + Vab-c + */abc 2 - /abc(Ja + V6 + Vc) = the same equation may be expressed thus : Thus we have the biquadratic equation one of the roots of which is y = v /a + Jb + *Jc, while a, b, c are the roots of the cubic equation z 3 + P 2 + Qz R = 0. 98. In order to apply this resolution to the proposed equation y 4 +py 2 + qy + r = Q, we must express the assumed coefficients P, Q, II by means of p, q, r, the coefficients of that equation. For this purpose, let us compare the equa tions and it immediately appears that 2P=/>, -8JK = q, P 2 -4Q = r and from these equations we find T> P n P 2 ~ ^ r T? _ _2! ~2 ^~ 16 ~64 Hence it follows that the roots of the proposed equation ire generally expressed by the formula vhere a, b, c denote the roots of this cubic equation, But to find each particular root, we must consider, that as the square root of a number may be either positive or nega tive, so each of the quantities Ja, /&, ,Jc may have either the sign + or - prefixed to it; and hence our formula will give eight different expressions for the root. It is, however, to be observed, that as the product of the three quantities ^fa, */&, Jc must be equal to ^/ll or to - 1 ; when q is positive, their product must be a negative 8 quantity, and this can only be effected by making either one or three of them negative; again, when q is negative, their product must be a positive quantity ; so that in this case they must either be all positive, or two of them must be negative. These considerations enable us to determine, that four of the eight expressions for the root belong to the case in which q is positive, and the other four to that in which it is negative. 99. We shall now give the result of the preceding in vestigation in the form of a practical rule; and as the coefficients of the cubic equation which has been found involve fractions, we shall transform it into another, iu v which the coefficients are integers, by supposing z = -. Thus the equation " :_- = <) (54 2 16 becomes, after reduction, v 3 + 2pv 2 + (p 2 - 4/ )w - q- = , it also follows, that if the roots of the latter equation are a, b, c, the roots of the former are - , , so that our rule may now be expressed thus: Let y^+pi/ + qy + r = be any biquadratic equation wanting its second term. Form this cubic equation t 3 + 2pv* + (jfi 4.r)v q- = , and find its roots, which, let us denote by a, b, c. Then the roots of the proposed biquadratic equation are. when q is negative, y when q is positive, 100. As an example of the method of resolving a bi quadratic equation, let it be required to determine the roots of the following, By comparing this equation with the general formula, we have p = - 25, q = + GO, r = - 30 ; hence 2p = - 50, p* 2 - 4r = TG9, q 2 = 3GOO , and the cubic equation to be resolved is 03 _50i> 2 + 769w- 3600 = 0; the roots of which are found, by the rules for cubies, to be 9, 16, and 25, so that v /a = 3, */& = 4, c = 5. Now in this case q is positive, therefore a;=i(_3-4-f>)= -6, aj={(-3+4+5)=+3, a;=|(+3+4-5)=+l. 101. We have now explained the particular rules by which the roots of equations belonging to each of the first four orders may be determined; and this is the greatest length mathematicians have been able to go in the direct resolution of equations; for as to those of the fifth, and all higher degrees, no general method has hitherto been found, either for resolving them directly, or reducing them tc others of an inferior degree. It even appears that the formulas which express the roots of cubic equations are not of universal application; for in one case, that is, when the roots are all real, they become illusory, so that no conclusion can be drawn from them. The same observation will also apply to the formulae for the roots of biquadratic equations, because, before they can be applied, it is always necessary to find the roots of a cubic equation. But both in cubics and in biquadratic equations, even when the formulae involve no imaginary quantities, and therefore can be always applied, it is more convenient in practice to employ other methods, which we
are hereafter to explain.
