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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 853 · 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 = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 5 (c) 9 (d) 1
  2. 2.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 13 (c) 11 (d) 12
  3. 3.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 73 (d) 17
  4. 4.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
  5. 5.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
  6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  7. 7.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 89 (c) 9 (d) 169
  8. 8.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)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10414 (c) 10404 (d) 10004
  10. 10.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 68 (c) 100 (d) 132
  11. 11.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6561 (c) 6401 (d) 6571
  12. 12.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 201 (c) 366 (d) 256
  13. 13.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 541 (d) 271
  14. 14.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 15 (c) 53 (d) 54
  15. 15.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  16. 16.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
  17. 17.If a + b = 7 and ab = 10, what is a² + b²? (a) 69 (b) 49 (c) 39 (d) 29
  18. 18.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 15 (d) 56
  19. 19.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 127 (c) 1 (d) 85
  20. 20.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 10609 (c) 9419 (d) 9409
  21. 21.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
  22. 22.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 9 (c) 3 (d) 17
  23. 23.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8464 (c) 7744 (d) 8474
  24. 24.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 352 (c) 160 (d) 256
  25. 25.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 1089 (c) 909 (d) 729
  26. 26.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 36 (c) 26 (d) 4
  27. 27.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 109 (c) 169 (d) 229
  28. 28.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 225 (c) 333 (d) 171
  29. 29.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 193 (d) 361
  30. 30.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
  31. 31.If a + b = 14 and ab = 45, what is (a − b)²? (a) 16 (b) 106 (c) 196 (d) 4
  32. 32.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 55 (d) 56
  33. 33.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 22 (c) 13 (d) 16
  34. 34.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 18 (d) 16
  35. 35.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 45 (c) 14 (d) 13
  36. 36.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2499 (c) 2601 (d) 2411
  37. 37.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
  38. 38.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 12) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  39. 39.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 273 (c) 361 (d) 185
  40. 40.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1235 (c) 1225 (d) 1575

Answer key

  1. 1
  2. 12
  3. 72
  4. (x + 4)(x + 7)
  5. (x + 3)(x + 6)
  6. (x + 8)(x + 9)
  7. 9
  8. (x + 3)(x + 9)
  9. 10404
  10. 68
  11. 6561
  12. 146
  13. 181
  14. 54
  15. (x + 5)(x + 7)
  16. (x + 6)(x + 8)
  17. 29
  18. 15
  19. 1
  20. 9409
  21. (x + 3)(x + 7)
  22. 9
  23. 8464
  24. 160
  25. 1089
  26. 16
  27. 109
  28. 117
  29. 193
  30. (x + 6)(x + 7)
  31. 16
  32. 56
  33. 10
  34. 17
  35. 14
  36. 2401
  37. (x + 7)(x + 9)
  38. (x + 2)(x + 6)
  39. 185
  40. 1225

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