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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 995 · 40 questions
TalentJR

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

NameDateTime takenScore ___ / 40
  1. 1.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 225 (c) 9 (d) 153
  2. 2.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 400 (d) 304
  3. 3.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181
  4. 4.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
  5. 5.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 48 (d) 14
  6. 6.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 2)(x + 9) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
  7. 7.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 10201 (c) 9801 (d) 9999
  8. 8.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 145 (d) 217
  9. 9.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 1 (c) 13 (d) 25
  10. 10.If a + b = 12 and ab = 11, what is (a − b)²? (a) 144 (b) 10 (c) 100 (d) 122
  11. 11.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  12. 12.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 2 and 28 (d) 7 and 8
  13. 13.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
  15. 15.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 10) (c) (x + 1)(x + 6) (d) (x + 2)(x + 5)
  16. 16.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 12 (d) 13
  17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  18. 18.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 12 (c) 11 (d) 10
  19. 19.If a + b = 9 and ab = 14, what is (a − b)²? (a) 25 (b) 81 (c) 5 (d) 53
  20. 20.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 6 (c) 8 (d) 7
  21. 21.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 10001 (c) 9801 (d) 10211
  22. 22.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
  23. 23.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
  24. 24.What are the roots of x² − 7x + 10 = 0? (a) 2 and 5 (b) 1 and 10 (c) 1 and 6 (d) 3 and 4
  25. 25.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
  26. 26.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 129 (c) 89 (d) 169
  27. 27.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 94 (c) 49 (d) 64
  28. 28.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 9996 (c) 9604 (d) 10404
  29. 29.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1774 (c) 1764 (d) 1444
  30. 30.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 3 (c) 9 (d) 169
  31. 31.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 54 (c) 16 (d) 15
  32. 32.If a + b = 13 and ab = 12, what is (a − b)²? (a) 169 (b) 11 (c) 121 (d) 145
  33. 33.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8291 (c) 8101 (d) 7921
  34. 34.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4771 (d) 4761
  35. 35.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 90 (c) 117 (d) 198
  36. 36.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 3025 (d) 2475
  37. 37.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 81 (c) 45 (d) 9
  38. 38.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 9 (c) 185 (d) 361
  39. 39.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 7 (c) 13 (d) 9
  40. 40.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 100 (c) 91 (d) 82

Answer key

  1. 81
  2. 208
  3. 181
  4. (x + 5)(x + 6)
  5. 48
  6. (x + 2)(x + 9)
  7. 9801
  8. 145
  9. 1
  10. 100
  11. (x + 6)(x + 9)
  12. 7 and 8
  13. (x + 2)(x + 4)
  14. (x + 7)(x + 8)
  15. (x + 2)(x + 5)
  16. 13
  17. (x + 6)(x + 8)
  18. 11
  19. 25
  20. 7
  21. 10201
  22. 16
  23. (x + 4)(x + 6)
  24. 2 and 5
  25. (x + 2)(x + 6)
  26. 89
  27. 34
  28. 9604
  29. 1764
  30. 9
  31. 15
  32. 121
  33. 8281
  34. 4761
  35. 90
  36. 2025
  37. 9
  38. 9
  39. 5
  40. 82

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