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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 164 · 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 45²? (a) 2025 (b) 2035 (c) 2475 (d) 3025
  2. 2.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 1 and 36 (c) 2 and 18 (d) 4 and 9
  3. 3.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
  4. 4.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 2 and 36 (d) 1 and 16
  5. 5.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 2 and 20 (c) 1 and 40 (d) 5 and 8
  6. 6.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 219 (c) 429 (d) 289
  7. 7.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
  8. 8.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 333 (d) 441
  9. 9.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
  10. 10.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 17 (c) 33 (d) 25
  11. 11.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 20 (c) 1 and 8 (d) 4 and 5
  12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  13. 13.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 11 (c) 12 (d) 10
  14. 14.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 65 (d) 177
  15. 15.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 421 (c) 289 (d) 157
  16. 16.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 157 (c) 289 (d) 5
  17. 17.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 10 (c) 9 (d) 18
  18. 18.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  19. 19.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 11 (d) 13
  20. 20.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 576 (d) 586
  21. 21.What are the roots of x² − 11x + 18 = 0? (a) 3 and 6 (b) 1 and 10 (c) 2 and 9 (d) 1 and 18
  22. 22.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 4 (c) 7 (d) 6
  23. 23.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  24. 24.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 196 (c) 100 (d) 148
  25. 25.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
  26. 26.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 7 (c) 5 (d) 9
  27. 27.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 156 (c) 116 (d) 196
  28. 28.If a + b = 7 and ab = 10, what is (a − b)²? (a) 9 (b) 3 (c) 29 (d) 49
  29. 29.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 193 (c) 5 (d) 25
  30. 30.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 1 and 11 (c) 2 and 10 (d) 1 and 35
  31. 31.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4904 (c) 4624 (d) 5194
  32. 32.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 181 (d) 541
  33. 33.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1034 (c) 784 (d) 1024
  34. 34.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 229 (c) 109 (d) 139
  35. 35.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 14 (d) 13
  36. 36.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 202 (d) 2
  37. 37.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 366 (c) 201 (d) 146
  38. 38.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 916 (c) 1166 (d) 676
  39. 39.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1444 (c) 1774 (d) 1604
  40. 40.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4499 (c) 5329 (d) 4891

Answer key

  1. 2025
  2. 4 and 9
  3. 72
  4. 8 and 9
  5. 5 and 8
  6. 149
  7. (x + 4)(x + 6)
  8. 225
  9. 56
  10. 17
  11. 4 and 5
  12. (x + 8)(x + 9)
  13. 11
  14. 65
  15. 157
  16. 25
  17. 9
  18. (x + 6)(x + 7)
  19. 12
  20. 576
  21. 2 and 9
  22. 5
  23. (x + 7)(x + 9)
  24. 100
  25. (x + 6)(x + 8)
  26. 1
  27. 116
  28. 9
  29. 25
  30. 5 and 7
  31. 5184
  32. 181
  33. 1024
  34. 109
  35. 13
  36. 4
  37. 146
  38. 1156
  39. 1764
  40. 4489

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