2. If there are two radicals and other terms, make the more complex radical constitute one member, alone, before squaring. Such cases usually require two involutions. 3. If there is a radical denominator, and radicals of a similar form occur in the numerators or constitute other terms, it may be best to clear of fractions first, either in whole or part. 4. It is sometimes best to rationalize a radical denominator. (3bc+ad)x 5ab (3bc-ad)x 5a(2b-a) = 2ab(a - b) 5m(g3-2x). (x 8.5 .2 х 4 = 41 x= 1 .1x X a2-b2 Sab2+4b3-12a2b m 12986). (6.) 3a2+ub-ac + bc 3c-d 4m (K-5x2) 8x e 9 + + (4 – 5) 4 3. In solving the following be careful to observe the suggestions in (27): _ (1) }(x-3)—}(x−¦) +1 (x−§) = 0. (2). X 6x-4 3 2 4. Solve the following, giving special heed to the suggestions in (28): (1.) √x−32 = 16 −√√x. (2.) =C. (3.) √x √b2+x-b +√a−√x=√x. (6.) √(1+a)2+(1−a)x+√(1−—a)2+(1+a)x =2a. (7.) √ {13+√ [7+√(3+√x)]}=4. (8.) √1+√(3+√/6x) [Several of these equations can be more elegantly reduced by the method given on p. 138, Ex. 47.] APPLICATIONS. 29. According to the definition (2), Algebra treats of, 1st, The nature and properties of the Equation; and 2d, the method of using it as an instrument for mathematical investigation. Having on the preceding pages explained the nature and properties of the equation, we now give a few examples to illustrate its utility as an instrument for mathematical investigation. 30. The Algebraic Solution of a problem consists of two parts: 1st. The Statement, which consists in expressing by one or more equations the conditions of the problem. 2d. The Solution of these equations so as to find the values of the unknown quantities in known ones. This process has been explained, in the case of Simple Equations, in the preceding articles. 31. The Statement of a problem requires some knowledge of the subject about which the question is asked. Often it requires a great deal of this kind of knowledge in order to "state a problem." This is not Algebra; but it is knowledge which it is more or less important to have according to the nature of the subject. 32. Directions to guide the student in the Statement of Problems: 1st. Study the meaning of the problem, so that, if you had the answer given, you could prove it, noticing carefully just what operations you would have to perform upon the answer in proving. This is called, Discovering the relations between the quantities involved. 2d. Represent the unknown (required) quantities (the answer) by some one or more of the final letters of the alphabet, as x, y, z, or w, and the known quantities by the other letters, or, as given in the problem. 3d. Lastly, by combining the quantities involved, both known and unknown according to the conditions given in the problem (as you would to prove it, if the answer were known) express these relations in the form of an equation. 33. SCH.-It is not always expedient to use a to represent the number sought. The solution is often simplified by letting a be taken for some number from which the one sought is readily found, or by letting 2x, 3x, or some multiple of a stand for the unknown quantity. The latter expedient is often used to avoid fractions. PROBLEMS. 1. A's age is double B's, B's is triple C's, and the sum of their ages is 140. Required the age of each. 2. A's age is m times B's, B's is n times C's, and the sum of their ages is s. Required the age of each. 3. The sum of two numbers is 48, and their difference 12. What are the numbers? 4. The sum of two numbers is s, and their difference d. What are the numbers? 5. Having the sum and difference of two numbers given, how do you find the numbers, arithmetically? 6. A post is th in the earth, ths in the water, and 13 feet in the air. What is the length of the post? n 8 m 7. A post is 1th in the earth, ths in the water, and a feet in the air. What is the length of the post? 8. What fraction is that, whose numerator is less by 3 than its denominator; and if 3 be taken from the numerator, the value of the fraction will be ? 9. Give the general solution of the last; i. e., the solution when the numbers are all represented by letters. Then substitute the above numbers and find the answer to that special problem. m n SUG.-Letting the numerator be a less than the denominator, and be the am + bn fraction after b is taken from the numerator, the fraction is an + bn' 10. A man sold a horse and chaise for $200; the price of the horse was equal to the price of the chaise. Required, the price of each. Chaise, $120; horse, $80. Generalize and solve the last, and then by substituting the numbers given in it find the special answers. Treat in like manner the next nine problems. 11. Out of a cask of wine which had leaked away a third part, 21 gallons were afterward drawn, when it was found that one-half remained. How much did the cask hold? Ans., 126 galls. 12. A and B can do a piece of work in 12 days, but when A worked alone he did the same work in 20. How long would it take B to do the same work? Ans., 30 days. L 13. A cistern can be filled by 3 pipes; by the first in 14 hours, by the second in 24 hours, and by the third in 5 hours. In what time will the cistern be filled, when all are left open at once? 14. Four merchants entered into a speculation, for which they subscribed 4755 dollars; of which B paid three times as much as A; C paid as much as A and B; and D paid as much as C and B. What did each pay? 15. A and B trade with equal stocks. In the first year A tripled his stock and had $27 to spare; B doubled his stock, and had $153 to spare. Now the amount of both their gains was five times the stock of either. What was that? 46 16. A and B began to trade with equal sums of money. In the first year A gained 40 dollars, and B lost 40; but in the second A lost one-third of what he then had, and B gained a sum less by 40 dollars than twice the sum that A had lost; when it appeared that B had twice as much money as A. What money did each begin with? д Ans., 320 dollars. 17. What number is that to which if 1, 5, and 13 be severally added, the first sum divided by the second shall equal the second divided by the third? 18. Divide 49 into two such parts that the greater increased by 62 divided by the less diminished by 11, shall be 41. 19. A cistern which contains 2400 gallons can be filled in 15 minutes by three pipes, the first of which lets in 10 gallons per minute, and the second 4 gallons less than the third. How much passes through each pipe in a minute? 20. Find a number such that, if from the quotient of the number increased by 5, divided by the number increased by 1, we subtract the quotient of 3 diminished by the number, divided by the number diminished by 2, the remainder shall be 2. 21. Divide a into two such parts, that one may be the th part of the other. 22. Divide a into two such parts, that the sum of the quotients which are obtained by dividing one part by m, and the other by n, m(nb-a) n(mb-a) shall be equal to b. The parts are n-m and m-n |