And x = a-I b 62—2bc+ca ao 22 Fi. Sometimes different powers of the unknown quantity are found in the equation, yet the several terms may form on one side a perfect power, of which the root being extracted, the equation will become simple. Thus, if x?-12x2+48x=98. It is easy to observe that *--12x2+48x—64=34, forming a complete cube, of which the root being extracted, *~4=\34. And x=4 + 34. EXAMPLE I. 1 To find proportionals, of which the sum of the extremes is 56, and the sum of the means 24. To resolve the question in general terms, let the sum of the extremes be a, the sum 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 als are 2 The proportion- ratzb+y. by.. 12 2 2 Mult. by 2 and 2a +2 : b+y:b-y:a-2 still From the three 3'ab-ay+bz-zy=b3+2by+y?.. first From the three fab+ay--bz-zy=b2-2by+y? last 3d added to 4th 52ab-—2zy=262 +2y? 4th subtr. from 62b2--2ay=4by 3d ( 6th reduced 72 7th subst. for a 82ab2 * 6 2 galo_63=3by+ay? ab2-63 3b+a amb In numbers 11 y=M 3 bata 36+a240 in sth 2 IO 32 128= Hence the four proportionals are 54, 18, 6, 2; and it appears that B must not be greater than a, otherwise the root becomes imposible, and the problem would also be impossible; which limitation might be deduced also from Prop. 25. V. of Euclid. II. Solution of adfected Quadratic Equa, tions, Adfected equations of different orders An adfected quadratic equation (com- be resolved by the following RULE. ! 1. Transpose all the terms involving the un known quantity to one side, and the known quantity may be positive. multiplied by any coefficient, all the terms known quantity may be 1. and and the side of the equation involving the unknown quantity will be a complete Square. 4. Extract the square root from both sides of the equation, by which it becomes fimple, and by transposing the above mentioned half coefficient, a value of the unknown quantity is obtained in known terms, and therefore the equation is resolved, The reason of this rule is manifest from the composition of the square of a binomial, for it consists of the squares 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, expressed in general terms, being reduced by the first and second parts of the rule, are these: x2 + ax=62 -ax=82 Fax= 3. *? -62 =+ B2+1 ?-ax=62 a? 4 2 a H 4 Case 3. ax=-62 a aa ax + -62 4 Van *== 62. Of these cases it may be observed, 1. That if it be supposed, that the square root of a positive quantity may be either positive or negative, according to the most extensive use of the signs, every quadratic equation will have two roots, except such 0 of |