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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 215 · 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.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 7)
  2. 2.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
  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 = 12 and ab = 20, what is a² + b²? (a) 184 (b) 104 (c) 144 (d) 124
  5. 5.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 891 (c) 739 (d) 1089
  6. 6.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 202 (d) 400
  7. 7.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 1 and 35 (c) 5 and 7 (d) 2 and 10
  8. 8.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 25 (d) 23
  9. 9.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3731 (c) 3721 (d) 3601
  10. 10.If a + b = 11 and ab = 18, what is (a − b)²? (a) 49 (b) 121 (c) 85 (d) 7
  11. 11.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 214 (c) 144 (d) 74
  12. 12.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 5329 (c) 4489 (d) 4891
  13. 13.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 121 (c) 157 (d) 85
  14. 14.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
  15. 15.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 9801 (c) 10211 (d) 10201
  16. 16.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 169 (c) 85 (d) 253
  17. 17.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x + 1)(x + 12)
  18. 18.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 451 (c) 441 (d) 401
  19. 19.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 9 (c) 8 (d) 10
  20. 20.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 148 (c) 100 (d) 292
  21. 21.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 56 (d) 57
  22. 22.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 7754 (c) 8096 (d) 8464
  23. 23.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 1 (c) 25 (d) 49
  24. 24.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
  25. 25.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 12 (c) 13 (d) 27
  26. 26.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 19 (d) 20
  27. 27.If a + b = 9 and ab = 8, what is (a − b)²? (a) 49 (b) 81 (c) 65 (d) 7
  28. 28.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 541 (d) 271
  29. 29.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 48 (d) 14
  30. 30.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 2 and 27 (c) 1 and 54 (d) 6 and 9
  31. 31.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 256 (c) 416 (d) 586
  32. 32.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 8 and 9 (d) 1 and 16
  33. 33.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 576 (d) 256
  34. 34.If a + b = 13 and ab = 30, what is (a − b)²? (a) 7 (b) 169 (c) 109 (d) 49
  35. 35.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
  36. 36.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 1 (c) 7 (d) 9
  37. 37.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 100 (c) 142 (d) 58
  38. 38.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 175 (c) 225 (d) 325
  39. 39.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 177 (c) 93 (d) 65
  40. 40.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)

Answer key

  1. (x + 2)(x + 6)
  2. (x + 3)(x + 4)
  3. (x + 8)(x + 9)
  4. 104
  5. 729
  6. 202
  7. 5 and 7
  8. 24
  9. 3721
  10. 49
  11. 74
  12. 4489
  13. 85
  14. 16
  15. 10201
  16. 85
  17. (x + 4)(x + 9)
  18. 441
  19. 9
  20. 100
  21. 56
  22. 7744
  23. 1
  24. (x + 2)(x + 4)
  25. 12
  26. 20
  27. 49
  28. 181
  29. 48
  30. 6 and 9
  31. 576
  32. 8 and 9
  33. 256
  34. 49
  35. (x + 6)(x + 7)
  36. 1
  37. 58
  38. 125
  39. 65
  40. (x + 5)(x + 8)

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