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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 573 · 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 = 7 and ab = 12, what is a² + b²? (a) 73 (b) 25 (c) 37 (d) 49
  2. 2.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 9604 (c) 10404 (d) 10414
  3. 3.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 7235 (c) 9025 (d) 8075
  4. 4.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 10 (c) 23 (d) 24
  5. 5.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 1)(x + 40)
  6. 6.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 193 (d) 361
  7. 7.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 90 (c) 198 (d) 144
  8. 8.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
  9. 9.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
  10. 10.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 9 (c) 11 (d) 21
  11. 11.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
  12. 12.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 142 (c) 79 (d) 58
  13. 13.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)
  14. 14.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
  15. 15.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  16. 16.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 4499 (c) 4489 (d) 5329
  17. 17.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)
  18. 18.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 9 (c) 17 (d) 3
  19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
  20. 20.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1616 (c) 1936 (d) 1296
  21. 21.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
  22. 22.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1575 (c) 1235 (d) 1225
  23. 23.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3625 (c) 4225 (d) 3025
  24. 24.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 794 (c) 896 (d) 784
  25. 25.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
  26. 26.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 189 (c) 153 (d) 297
  27. 27.If a + b = 5 and ab = 6, what is a² + b²? (a) 13 (b) 37 (c) 19 (d) 25
  28. 28.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 48 (c) 13 (d) 15
  29. 29.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 48 (d) 14
  30. 30.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3146 (c) 3136 (d) 4096
  31. 31.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2304 (c) 2496 (d) 2704
  32. 32.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4916 (c) 4356 (d) 5476
  33. 33.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 2 (c) 4 (d) 196
  34. 34.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 9) (d) (x + 1)(x + 16)
  35. 35.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 16 (d) 4
  36. 36.If a + b = 8 and ab = 12, what is (a − b)²? (a) 16 (b) 40 (c) 4 (d) 64
  37. 37.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 196 (c) 163 (d) 130
  38. 38.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 196 (c) 8 (d) 64
  39. 39.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 256 (c) 376 (d) 196
  40. 40.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 4896 (c) 5184 (d) 4634

Answer key

  1. 25
  2. 10404
  3. 7225
  4. 24
  5. (x + 5)(x + 8)
  6. 193
  7. 90
  8. (x + 3)(x + 9)
  9. (x + 3)(x + 6)
  10. 10
  11. (x + 5)(x + 9)
  12. 58
  13. (x + 8)(x + 9)
  14. (x + 6)(x + 9)
  15. (x + 6)(x + 8)
  16. 4489
  17. (x + 6)(x + 7)
  18. 9
  19. (x + 7)(x + 9)
  20. 1936
  21. (x + 3)(x + 7)
  22. 1225
  23. 4225
  24. 784
  25. 17
  26. 153
  27. 13
  28. 14
  29. 48
  30. 3136
  31. 2304
  32. 5476
  33. 4
  34. (x + 2)(x + 8)
  35. 16
  36. 16
  37. 130
  38. 64
  39. 136
  40. 4624

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