Sometimes different powers of the unknown quantity are found in the equation, yet the feveral terms may form on one fide a perfect power, of which the root being extracted, the equation will become fimple. Thus, if x3—12x2+48x=98. It is eafy to observe that x-12x2+48x-64-34, forming a complete cube, of which the root being extracted, x-4=3√34. And x=4 +3√34. EXAMPLE I. To find four continued proportionals, of which the fum of the extremes is 56, and the fum of the means 24. To refolve the question in general terms, let the fum of the extremes be a, the fum of the means b, and let the difference of the extremes be called z, and the difference of the means y. Then, by Ex. 8. Chap. 3. The The proportion + r'a+z b+y, b—y az als are Mult. by 2 and ftill From the three firft From the three laft 2 2 2a+z:b+y:b—y: a 2 3'ab-ay+bz-zy=b2+2by+y2 ·· 4 abay—bz―zy—b2—2by+y2 3d added to 4th 5 2ab—2zy=2b2+2y2 4th fubtr. from 3d 6th reduced 7th fubft. for x in 5th Tranf. and di 62bz-2ay=4by 72 - 269 +ay 82ab-2 X 2 by2 + ay2 = 2b2 + 2y2 9 ab2—b3=3by2+ay2 ΙΟ =y, and y = 36+a 36+a Hence the four proportionals are 54, 18, 6, 2; and it appears that B must not be greater than a, otherwise the root becomes -impoffible, and the problem would also be impoffible; which limitation might be deduced alfo from Prop. 25. V. of Euclid. II. Solution of adfected Quadratic Equa tions. Adfected equations of different orders are refolved by different rules, fucceffively to be explained. An adfected quadratic equation (commonly called a quadratic) involves the unknown quantity itself, and alfo its fquare: may be refolved by the following It RULE. 1. Tranfpofe all the terms involving the unknown quantity to one fide, and the known terms to the other; and fo that the term containing the Square of the unknown quantity may be pofitive. 2. If the fquare of the unknown quantity is multiplied by any coefficient, all the terms of the equation are to be divided by it, fo that the coefficient of the fquare of the known quantity may be 1. 3. Add to both fides the fquare of half the coefficient of the unknown quantity itself, and and the fide of the equation involving the unknown quantity will be a complete Square. 4. Extract the fquare root from both fides of the equation, by which it becomes fimple, and by tranfpofing the above mentioned half coefficient, a value of the unknown quantity is obtained in known terms, and therefore the equation is refolved, The reafon of this rule is manifeft from the compofition of the fquare of a binomial, for it confifts of the fquares of the two parts, and twice the product of the two parts. (Note at the end of Chap. IV.), The different forms of quadratic equations, expreffed in general terms, being reduced by the firft and second parts of the rule, are these : |