the 6th order, but of the quadratic form, is deduced, which gives the values of m and n, and hence, m+n=x=`√=r+√r2+q3+ 3 √√r2+q3 ; Cor. I. In the given equation, if 39 is negative, and if 2 is lefs than q, this expreffion of the root involves impoffible quantities; while, at the fame time, all the roots of that equation are poffible. The reafon is, that, in this method of folution, it is neceffary to suppose that x the root may divided into two parts, of which the product is q. But it is easy to shew, that, in this, which is called the irreducible case, it cannot be done. be For example, the equation, (Ex. 3. Sect. 3. of this chapter), x-156x+560=0, belongs to the irreducible cafe, and the three roots are +4, +10, -14, and it is plain that none of these roots can be divided into two parts m and n, of which the product can be equal to -q= 156 =52; for the 3 greatest greatest product from the division of the greatest root-14, is −7X-7=49, less than 52. If the cube root of the compound furd can be extracted, the impoffible parts balance each other, and the true root is obtained. The geometrical prob'em of the trifection of an arch is refolved algebraically, by a cubic equation of this form ; and hence the foundation of the rule for refolving an equation belonging to this cafe, by a table of fines. Cor. 2. Biquadratic equations may be reduced to cubics, and may therefore be refolved by this rule. Some other claffes of equations too, may be refolved by particular rules; but thefe, and every other order of equations, are commonly refolved by the general rules, which may be equally applied to all. II. II. Solution of Equations, whofe Roots are commenfurate. All the terms of the equation being brought to one fide, find all the divifors of the abfolute term, and fubflitute them fucceffively in the equation for the unknown quantity. That divifor which, fubftituted in this manner, gives the refult =0, fhall be a root of the equation. . -2a2b} Ex. 1. x3-3x2+2a2x-2a2b =0. The fimple literal divifors of -2a1b, are a, b, 2a, 2b, any of which may be inferted for x. becomes Suppofing x=+a, the equation a2b} which is obviously=0% a3—3a3+2a3—2a2b -ba2+3a2b Ex. 2. x3-2x2-33x+90=0. The divifors of 90 are 1, 2, 3, 5, 6, 9; 10, 15, 18, 30, 45, 90% The first of these divifors, which being inferted for x, will make the result =o, is +3; +3; +5 is another, and it is plain the last root must be negative, and it is -6. When 3 is difcovered to be a root, the given equation may be divided by x-3 =0, and the refult will be a quadratic, which being refolved, will give the other two roots, +5 and -6. The reason of the rule appears from the property of the abfolute term formerly defined, viz. that it is the product of all the roots. To avoid the inconvenience of trying many divisors, this method is fhortened by the following. RULE II. Subftitute in place of the unknown quantity fucceffively three or more terms of the progreffion, 1, 0,-1, &c. and find all the divifors of the fums that refult; then take out all the arithmetical progressions that can be found among these divifors whofe common aifference is 1, and the values of x will be among those terms of the progreffions which are the divifors of the result arifing from the fubftitution of x=0. When the feries increafes, the roots will be pofitive; and when it decreases, the roots will be negative. EXAMPLE. Let it be required to find a root of the equa tion x-x2-10x+6=0. In this example there is only one progreffion, 4, 3, 2, and therefore 3 is a root, and it is -3, fince the series decreases. It is evident from the rules for transforming equations, (Chap. 2.), that by inserting for x, +1(=+e) the result is the absolute term of an equation, of which the roots are lefs |