be found, so that y may be an integer; and, in the fourth, x is to be found, fo that y may be a rational quantity. a 2. When y=x, it is plain that, in b+cx order to have y an integer, b+cx must be a divifor of a. Let d be one of its divi fors, then, if b+cx=d, x=db; so that C among the divifors of a, we must find one, if poffible, from which b being taken away, the remainder may be divisible by c: the quotient is a value of x. 3. When a+bx y= <+dx' if d be a divifor of b, x will be taken out of the numerator, if we divide it by dx+c; and this form is then reduced to the preceding. But, if d is not a divifor of b, multiply both sides by d, then dy ad bdx or, dividing bdx+ad by by dx+c, dy=b+ad-be, and fox is found c+dx by making c+dx equal to a divifor of ad-bc. Example. Let 2xy+x+y=195; then y (2x+1)= and if 1+2x=17, x=8, and y = 11. 4. When y=√a+bx+cx2, and x is to be found, so that y may be a rational quantity. Here there are four cases, according to the nature of the coefficients a, b, and c. Imo, If a be a square number, as for instance g2, so that the formula is √2+bx+cx2. Suppose √g2+bx+cx2 = g+mx, then g2+bx+cx2=g2+2mgx+m2x2, or bx+ cx2 = 2mgx+m2x2, that is, b+cx=2mg 2mg-b +m2x, and x= If for x this value of it be substituted in the given formula, its irrationality will dif appear, and g2+bx+cx2 = cg—bm+gm2 ; c-m2 m may be affumed, therefore, equal to any quantity, pofitive or negative, integral or fractional, and the correfponding value of x will answer the conditions prescribed. 2do, If c be a fquare number, as g2, then let a+bx+g2x2=m+gx. Hence a+bx+g2x2=m2+2mgx+g2x2, or a+bx =m2+2mgx. Therefore x = m--a and b-2mg bm-gm2-ag. Here m, as √a+bx+g2x2 b-2mg before, may be affumed at pleasure. 3tio, Though neither a nor c are square numbers, yet if a+bx+cx2 can be resolved into two fimple factors, as f+gx, and bkx, the irrationality of the formula may be taken away. For, let a+bx+cx2 = √(f+gx) (b+kx)=m(f+gx), and (f+gx) (b+kx)=m2(f+gx)2, or b+kx = 2 =m2 (f+gx), and x= _fm2-b k--gm2 By the sub flitution of this value of x, (m being affu med med at pleasure), the irrationality will be removed, as before. 4to, The fourth cafe, in which the formùla a+bx+cx2 may be rendered a complete fquare, is when it can be divided into two parts, one of which is a complete fquare, and the other a produd of two fimple factors. For a+bx+cx2 is then of the form p2-qr, p, q, and r, being quantities into which there enters no power of x higher than the firft: And, if we affume √p2+qr=p+mq, p2 will be exterminated, and the remainder, being divided by 9, will be a fimple equation, from which x may be easily determined. These methods of removing the irrationality of the preceding formula are to be particularly attended to, as being of great ufe in the higher geometry. APPEN |