1. 4(x − 1) (x − 2) (x − 3) — 3 (x − 2) (x −3). 1, 2, 3, 4. α. r=1, s= -1; r=2, s=-2; ra, s=-a. 11. a+b+c=6. a=1, b=2, c=3; a=3, b=3, c=0; a=10, b=0, c=-4. 13. a=8, b=0, c=6; a=1, b = 4, c = 2; a=0, b=0, c=-4. 14. (x — z) (x − y) (y − z) = 8 xyz (x2 — y2) (y2 — z2) (≈2 — x2). x=1,y=1, z=1; x=1, y=0, z=1; x=1, y=2, z=3. 15. 2+3x2+3xy2+y3 = (x+y)3. x=1, (x=1, fx=3, [y=1. │y=2. \y=4. 16. Show by reducing the equality in Ex. 15 to the form R= 0 that it is satisfied by any pair of values whatsoever for x and y, e.g., for x=348764, y=594021. What kind of an equality is this? Which of the following four equalities are identities? 17. 12(x+y)2+17 (x + y) −7 = (3x+3 y−1)(4x+4y+7). 20. 2(a - b)2+5(a + b) + 8 ab = (2a+2b+1)(a+b+1). Solve the following equations, and verify the results: 21. (2a+3)(3 a − 2) = a2 + a (5 a +3). 22. 6(6-4)-5-(3-2b)2-5 (2+b) (7 — 2b). 23. (y-3)2+(y — 4)2 — (y — 2)2 — (y — 3)2 — 0. = 24. (x-3) (3x+4) − (x-4) (x-2)=(2x+1)(x-6). 25. 2(3r-2)(4 r + 1) + (r − 4)3 — (r + 4)3 — 2. = 26. a3-c+b3c + abc-b. (Solve for c.) 27. (b−2)2 (b—y) − 3 by +(2b+1)(b−1)=3—2b. (Find y.) 28. 2(12-x)+3(5 x −4) +2 (16 − x) = 12 (3 + x). 29. (ba) x − (a + b) x + 4 a2 =0. (Find x.) 30. (x-a) (b-c)+(b− a) (x−c)-(a–c) (x—b)=0. 31. r3v+s3v-3 r−3s+3 v(r2s+rs2) = 0. (Find x.) (Find v.) 32. (x-3)(x-7) — (x − 5) (x − 2)+12=2(x-1). 33. (a+b)2 + (x − b) (x − a) — (x + a) (x + b) = 0. (Find x.) 34. ny(y+n)-(y+m) (y+n) (m+n)+my(y+m)=0. (Find y.) 35. (n+i)(j−i + k) − (n − i) (i − j + k) = 0. (Find n.) 16 36. fz(5x−1)+rỗ (2−3x)+}(4+)=g(1+2x). 37. (lm) (z - n) + 2 1 (m + n) = (l+m) (z + n). (Find z.) 38. a(x-b)-(a+b)(x+b− a)=b(x-a)+a2-b2. (Find x.) 39. (m+n)(n+ b − y) + (n − m)(b−y)=n(m+b). (Find y.) 40. 3 (2a-3b) 2 (3a-5b) 5 (a−b)_b. + 6 (Find a.) 8 Solve each of the following equations for each letter in terms of the others. 1. What number must be added to each of the numbers 2, 26, 10 in order that the product of the first two sums may equal the square of the last sum ? 2. What number must be subtracted from each of the numbers 9, 12, 18 in order that the product of the first two remainders may equal the square of the last remainder ? 3. What number must be added to each of the numbers a, b, c in order that the product of the first two sums may equal the square of the last? Note that problem 1 is a special case of 3. Explain how 2 may also be made a special case of 3. 4. What number must be added to each of the numbers a, b, c, d in order that the product of the first two sums may equal the product of the last two? 5. State and solve a problem which is a special case of problem 4. 6. What number must be added to each of the numbers a, b, c, d in order that the sum of the squares of the first two sums may equal the sum of the squares of the last two? 7. State and solve a problem which is a special case of problem 6. 8. What number must be added to each of the numbers a, b, c, d in order that the sum of the squares of the first two sums may be k more than twice the product of the last two? 9. State and solve a problem which is a special case of problem 8. 10. The radius of a circle is increased by 3 feet, thereby increasing the area of the circle by 50 square feet. Find the radius of the original circle. The area of a circle is πr2. Use 34 for π. 11. The radius of a circle is decreased by 2 feet, thereby decreasing the area by 36 square feet. Find the radius of the original circle. 12. State and solve a general problem of which 10 is a special case. 13. State and solve a general problem of which 11 is a special case. How may the problem stated under 12 be interpreted so as to include the one given under 13? 14. Each side of a square is increased by a feet, thereby increasing its area by b square feet. Find the side of the original square. Interpret this problem if a and b are both negative numbers. 15. State and solve a problem which is a special case of 14, (1) when a and b are both positive, (2) when a and b are both negative. 16. Two opposite sides of a square are each increased by a feet and the other two by b feet, thereby producing a rectangle whose area is e square feet greater than that of the square. Find the side of the square. Interpret this problem when a, b, and c are all negative numbers. 17. State and solve a problem which is a special case of 16, (1) when a, b, and c are all positive, (2) when a, b, and c are all negative. 18. A messenger starts for a distant point at 4 A.M., going 5 miles per hour. Four hours later another starts from the same place, going in the same direction at the rate of 9 miles per hour. When will they be together? When will they be 8 miles apart? How far apart will they be at 2 P.M.? For a general explanation of problems on motion, see p. 115, E. C. 19. One object moves with a velocity of v1 feet per second and another along the same path in the same direction with a velocity of 2 feet. If they start together, how long will it require the latter to gain n feet on the former? From formula (2), p. 117, E. C., we have t = n V2-1 Discussion. If v2>v, and n>0, the value of t is positive, i.e. the objects will be in the required position some time after the time of starting. If v2 < v1 and n > 0, the value of t is negative, which may be taken to mean that if the objects had been moving before the instant taken in the problem as the time of starting, then they would have been in the required position some time earlier. If If v2 = v1 and n‡0, the solution is impossible. See § 25. This means that the objects will never be in the required position. V1 = v2 and n = 0, the solution is indeterminate. See § 24. This may be interpreted to mean that the objects are always in the required position. 20. State and solve a problem which is a special case of 19 under each of the conditions mentioned in the discussion. 21. At what time after 5 o'clock are the hands of a clock first in a straight line? |