· EXAMPLES. 1. Let the quadratic equation x2 ·8x+15=0 be given; it is required to take away its second term. Suppose xy +4(y+3); 2. Let the equation x3-9x2+26x-34―0 be given; it is required to exterminate its second term. Thus the roots of the equation x4-x3-19x2+49x-30-o are +1, +2, +3, -5; but, by changing only the second and fourth terms, the equation becomes x4+x3-19x2-49x-30=0, and the roots are -1, -2, −3, +5. All the roots of an equation may also be made affirmative or ne gative, by increasing or diminishing each of them by some known quantity. * From this example it appears, that any quadratic equation may be solved without completing the square, by only taking away the second term; for since y2=1, or y=1=1, we shall have xy+ 4-1+4-5, the root required. And the same may be shown of any other adfected quadratic equation whatever. L 3. Let x48x3-5x2+10x-4=0 be given; to exterminate the second term*. Suppose xy —2(y-3); Then x4y48y3+2442-32y+16 4. Let x4x3+qx2--rx+s=0 be given; to exter * Since the sum of all the roots, in any equation, is equal to the co-efficient of the second term, it follows that, when the second term is wanting, the equation has both affirmative and negative roots, and that the sum of the affirmative roots is equal to the sum of the negative ones. Thus, in the cubic equation x3-7x=6, the three roots are +3, -2, and -1, where it is evident that 3=2+1. PROBLEM III. To find whether some or all the roots of an equation be rational; and, if so, what they are. RULE*. 1. Find all the divisors of the last term, and substitute them one by one for the unknown quantity. 2. Then, if the positive and negative terms destroy each other, the divisor, so substituted, will be one of the roots of the equation. 3. But if none of the divisors succeed, the roots are, for the general part, either irrational or impossible, Note. When the divisors of the last term are too numerous, they may be diminished by changing the equa tion into another, whose roots are augmented or decreased by a unit, or some other known quantity. - EXAMPLES. 1. Let x3. - 4x2 — 7x+10=0 be the equation proposed. Then the divisors of (10) the last term will be + 1, −1, +2, −2, +5, −5, +10, −10. * Since the last term, in any equation, is always equal to the product of all the roots in that equation, those roots must, therefore, necessarily be found in the number of its divisors. But this, it is evident, can hold only when the roots are commensurate, or whole numbers. And these, being substituted successively instead of x, Therefore +1,2, and +5 are the three roots of the equation required. 2. Let y4-443-8y+320 be the equation proposed. 1. Change it into another, the number of whose divisors shall be less; thus, xa—-6x2--16x+21=0=new equation*. 2. The divisors of the last term (21) of this new equation are 1, −1, +3, −3, +7, −7, +21, −21. And if these be substituted successively instead of x, we shall have 1-6-16+21=0 16+16+21=32 81-54-484-21= 0 81-54+48 +21=96 c. where none of the others succeed. So that 1 and 3 are the only rational roots, the other two being impossible. *Note. The divisor of the last term of this new equation may be diminished in the same manner as before. 3. Let x3+3ax2-4a2x-12a3=0, be the equation proposed. Here the numeral divisors of the last term (12a3) are }, −1, +2, −2, +3, −3, +4, −4, +6, -6, +12, -12. And by substituting these successively instead of x, we shall have 27+27-12-12= -27+27+12-12= Therefore the three roots are 2a, -2a, and -3a. PROBLEM IV. To discover the roots of equations by SIR ISAAC RULE. 1. Instead of the unknown quantity, substitute successively three, or more, terms of the arithmetical progression 2, 1, 0, −1, -2. 2. Collect all the terms of the equation into one sum, and place them, together with their divisors, in perpendicular lines, right against the corresponding terms of the progression 2, 1, 0, -1,` ——2. 3. Seek amongst the divisors for an arithmetical progression, whose terms correspond with the order of the terms 2, 1, 0, -1, -2, and whose common difference is either a unit, or some divisor of the co-efficient of the |