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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 703 · 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 = 16 and ab = 63, what is a² + b²? (a) 382 (b) 130 (c) 256 (d) 193
  2. 2.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4904 (c) 4624 (d) 5194
  3. 3.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
  4. 4.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 169 (d) 225
  5. 5.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 5 and 8 (c) 1 and 40 (d) 1 and 12
  6. 6.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 100 (c) 52 (d) 76
  7. 7.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
  8. 8.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 11)
  9. 9.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 42 (d) 12
  10. 10.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  11. 11.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
  12. 12.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 7225 (c) 9035 (d) 8125
  13. 13.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
  14. 14.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 400 (c) 304 (d) 208
  15. 15.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2209 (c) 2509 (d) 2809
  16. 16.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 181 (c) 61 (d) 91
  17. 17.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 11 (c) 10 (d) 12
  18. 18.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 925 (c) 625 (d) 1235
  19. 19.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
  20. 20.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 1 and 30 (d) 5 and 6
  21. 21.What is the product of the roots of x² − 12x + 27 = 0? (a) 26 (b) 12 (c) 27 (d) 28
  22. 22.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
  23. 23.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 121 (c) 97 (d) 73
  24. 24.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 247 (c) 478 (d) 170
  25. 25.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 5041 (d) 4761
  26. 26.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
  27. 27.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8091 (c) 8649 (d) 7569
  28. 28.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 433 (d) 145
  29. 29.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 586 (c) 256 (d) 416
  30. 30.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 4 (d) 16
  31. 31.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 196 (c) 148 (d) 100
  32. 32.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 225 (c) 9 (d) 441
  33. 33.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
  34. 34.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 69 (c) 39 (d) 49
  35. 35.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  36. 36.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 10 (d) 12
  37. 37.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1681 (d) 1601
  38. 38.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
  39. 39.If a + b = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 50 (c) 64 (d) 6
  40. 40.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 289 (c) 169 (d) 409

Answer key

  1. 130
  2. 5184
  3. (x + 3)(x + 9)
  4. 1
  5. 5 and 8
  6. 52
  7. (x + 3)(x + 4)
  8. (x + 4)(x + 8)
  9. 13
  10. (x + 7)(x + 9)
  11. (x + 2)(x + 3)
  12. 9025
  13. (x + 3)(x + 8)
  14. 208
  15. 2809
  16. 61
  17. 11
  18. 1225
  19. (x + 2)(x + 9)
  20. 5 and 6
  21. 27
  22. 181
  23. 73
  24. 170
  25. 5041
  26. 15
  27. 7569
  28. 145
  29. 576
  30. 16
  31. 100
  32. 9
  33. (x + 5)(x + 7)
  34. 29
  35. (x + 8)(x + 9)
  36. 11
  37. 1681
  38. (x + 4)(x + 5)
  39. 36
  40. 169

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