312. To find the condition that xo+px +q may be a perfect square. It must be evident that any such general expression cannot be a perfect square unless some particular relation subsists between the coefficients p and q. To find the necessary connection between p and q is the object of the present question. Using the ordinary rule for square root, we have 22+px+9(2+ р x2 px +9 + 2 + р 2x + p2 px + 4 p2 9 4 If therefore x2+px+q be a perfect square, the remainder, 9 must be zero. Hence the condition is determined by 4 placing this remainder equal to zero and solving the resulting equation. p2 EXAMPLES XXXIV. b. Write down the quotient in the following cases : w yo 1. 2. 3. 4. -Y -Y Resolve into factors : 5. 3 — 6x2 +11% — 6. 6. 23 – 5x2 - 2x -+24. 7. 23 +9x2 +26x +24. 8. 23 — 22 -41% +105. 9. 23 – 39x + 70. 10. 23 - 8x2 – 313 – 22. 11. 6x3 + 732 — 3-2. 12. 6x3 + x2 - 19x+6. Without actual division show that 13. 32x10 – 33x5 +1 is divisible by x–1. 14. 3x+ + 5x3 — 13x2 – 20x +4 .X2-4. 15. X+ + 4x3 — 5x2 – 36x - 36 .X2 - X --6. Without actual division find the remainder when 16. 26 – 5x2 +5 is divided by 2–5. p2 17. 203 — 732a +8xa+15a3 is divided by x+2a. 18. If n be any positive integer, prove that 52n – 1 is always divisible by 24. Find the values of x which will make each of the following expressions a perfect square : 19. 2+ +6x3+1322 + 13x – 1. 20. 2+ +6x3 +11x2 + 3x +31. 21. 2* — 2ax: +(a?+26) x2 – 3abx+262. 22. 4p224 — 4pq2c3+ (q2+2p2) x2 – 5pqx+ 2 2acx3 9b20c2 5bcx +6c2. 16 2 24. 24+ 2ax3 + 3a232 + cx+d. Find the values of x which will make each of the following expressions a perfect cube : 206 25. 8x3 -- 36x2 + 56x - 39. a2x4 26. +42422 - 28a6. 27 3 27. m8x8 - 9mnx4 +39mn2x2 – 51n3. + CHAPTER XXXV. RATIO, PROPORTION, AND VARIATION. 313. DEFINITION. Ratio is the relation which one quantity bears to another of the same kind, the comparison being made by considering what multiple, part, or parts, one quantity is of the other. The ratio of A to B is usually written A :B. The quantities A and B are called the terms of the ratio. The first term is called the antecedent, the second term the consequent. 314. To find what multiple or part A is of B we divide A by B; hence the ratio A :B may be measured by the fraction A and we shall usually find it convenient to adopt this notation. In order to compare two quantities they must be expressed in terms of the same unit. Thus the ratio of $2 to 15 cents is 2 x 100 40 measured by the fraction 15 3 or . Note. Since a ratio expresses the number of times that one quantity contains another, every ratio is an abstract quantity. a та 315. By Art. 120, bmb i and thus the ratio a : b is equal to the ratio ma : mb; that is, the value of a ratio remains unaltered if the antecedent and the consequent are multiplied or divided by the same quantity. 316. Two or more ratios may be compared by reducing their equivalent fractions to a common denominator. Thus suppose bx ay Now a : b and x : y are two ratios. and 7 by' y by the ratio a :b is greater than, equal to, or less than the ratio x : y according as ay is greater than, equal to, or less than bx. α ; hence 317. The ratio of two fractions can be expressed as a ratio of a c or ; a two integers. Thus the ratio is measured by the fraction ūã ad d bc and is therefore equivalent to the ratio ad : bc. 318. If either, or both, of the terms of a ratio be a surd quantity, then no two integers can be found which will exactly measure their ratio. Thus the ratio 12 :1 cannot be exactly expressed by any two integers. 4 and < 319. DEFINITION. If the ratio of any two quantities can be expressed exactly by the ratio of two integers the quantities are said to be commensurable; otherwise, they are said to be incommensurable. Although we cannot find two integers which will exactly measure the ratio of two incommensurable quantities, we can always find two integers whose ratio differs from that required by as small a quantity as we please. 15 2.236007... Thus •559016... 4 559017 and therefore is > 4 1000000 1000000" and it is evident that by carrying the decimals further, any degree of approximation may be arrived at. 320. DEFINITION. Ratios are compounded by multiplying together the fractions which denote them; or by multiplying together the antecedents for a new antecedent, and the consequents for a new consequent. Example. Find the ratio compounded of the three ratios 2a : 36, Cub : 5c", c:a. 2a Gab 4a The required ratio= 36 5c 5c x с Х a 321. DEFINITION. When the ratio a : b is compounded with itself the resulting ratio is a? : b2, and is called the duplicate ratio of a : b. Similarly a3 : 13 is called the triplicatē ratio of a :b. Also až : 82 is called the subduplicate ratio of a : b. 1 12 Examples. (1) The duplicate ratio of 2a : 36 is 4a: 962. (2) The subduplicate ratio of 49 : 25 is 7:5. (3). The triplicate ratio of 2x : 1 is 8x3 : 1. 322. DEFINITION. A ratio is said to be a ratio of greater inequality, of less inequality, or of equality, according as the antecedent is greater than, less than, or equal to the consequent. 323. If to each term of the ratio 8: 3 we add 4, a new ratio 12 : 7 is obtained, and we see that it is less than the former 8 because is clearly less than 7 3 This is a particular case of a more general proposition which we shall now prove. A ratio of greater inequality is diminished, and a ratio of less inequality is increased, by adding the same quantity to both its terms. a + x Let jy be the ratio, and let be the new ratio formed by adding a to both its terms. atx - bx Now +x6(6+2) x (a - ) a ar a 6+x; and a - b is positive or negative according as a is greater or less than 6. a+ Hence if a is >b, ū a atx and if a is <b, is < b b+x which proves the proposition. Similarly it can be proved that a ratio of greater inequality is increased, and a ratio of less inequality is diminished, by taking the same quantity from both its terms. 324. When two or more ratios are equal, many useful propositions may be proved by introducing a single symbol to denote each of the equal ratios. The proof of the following important theorem will illustrate the method of procedure. |