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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 144 · 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.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10609 (d) 10619
  2. 2.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 676 (c) 1166 (d) 916
  3. 3.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  4. 4.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 25 (c) 73 (d) 5
  5. 5.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 324 (c) 170 (d) 4
  6. 6.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
  7. 7.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 67 (c) 53 (d) 81
  8. 8.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3136 (c) 3146 (d) 4096
  9. 9.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3625 (c) 3025 (d) 4225
  10. 10.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 76 (c) 100 (d) 52
  11. 11.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 5776 (c) 6384 (d) 7056
  12. 12.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  13. 13.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 7056 (c) 6416 (d) 5776
  14. 14.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 33 (c) 17 (d) 25
  15. 15.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 4 (c) 106 (d) 16
  16. 16.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 91 (d) 61
  17. 17.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  18. 18.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 11)
  19. 19.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
  20. 20.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 5184 (c) 4634 (d) 4624
  21. 21.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 8 (c) 15 (d) 9
  22. 22.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
  23. 23.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 74 (c) 214 (d) 109
  24. 24.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 10 (c) 11 (d) 30
  25. 25.What is the product of the roots of x² − 8x + 15 = 0? (a) 14 (b) 15 (c) 16 (d) 8
  26. 26.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 196 (c) 10 (d) 148
  27. 27.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 9 (c) 10 (d) 18
  28. 28.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  29. 29.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 1 and 40 (c) 2 and 20 (d) 5 and 8
  30. 30.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 145 (c) 217 (d) 289
  31. 31.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
  32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 18 (d) 17
  33. 33.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1599 (c) 1681 (d) 1521
  34. 34.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3259 (c) 3969 (d) 3591
  35. 35.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
  36. 36.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 8099 (c) 7921 (d) 7931
  37. 37.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 36 (c) 20 (d) 28
  38. 38.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 132 (c) 84 (d) 68
  39. 39.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 1 (c) 127 (d) 85
  40. 40.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 4 (c) 2 (d) 256

Answer key

  1. 10609
  2. 1156
  3. (x + 8)(x + 9)
  4. 25
  5. 16
  6. (x + 3)(x + 4)
  7. 53
  8. 3136
  9. 4225
  10. 52
  11. 5776
  12. (x + 7)(x + 8)
  13. 7056
  14. 17
  15. 16
  16. 61
  17. (x + 5)(x + 8)
  18. (x + 5)(x + 7)
  19. (x + 4)(x + 9)
  20. 4624
  21. 8
  22. (x + 2)(x + 7)
  23. 74
  24. 11
  25. 15
  26. 100
  27. 9
  28. (x + 3)(x + 9)
  29. 5 and 8
  30. 145
  31. (x + 3)(x + 8)
  32. 17
  33. 1521
  34. 3249
  35. (x + 7)(x + 9)
  36. 7921
  37. 20
  38. 68
  39. 1
  40. 4

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