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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 200 · 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 39²? (a) 1531 (b) 1681 (c) 1521 (d) 1599
  2. 2.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  3. 3.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
  4. 4.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 49 (c) 14 (d) 47
  5. 5.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 193 (c) 361 (d) 25
  6. 6.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 29 (c) 3 (d) 9
  7. 7.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
  8. 8.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 13 (c) 14 (d) 40
  9. 9.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6241 (c) 6251 (d) 6399
  10. 10.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 3 (c) 9 (d) 225
  11. 11.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 484 (c) 334 (d) 396
  13. 13.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 145 (d) 289
  14. 14.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3136 (c) 4096 (d) 3616
  15. 15.If a + b = 8 and ab = 7, what is a² + b²? (a) 57 (b) 64 (c) 78 (d) 50
  16. 16.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 81 (c) 41 (d) 1
  17. 17.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
  18. 18.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 16 (c) 4 (d) 10
  19. 19.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 181 (d) 361
  20. 20.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
  21. 21.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10619 (c) 10009 (d) 10609
  22. 22.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6409 (c) 6889 (d) 6899
  23. 23.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 109 (c) 74 (d) 214
  24. 24.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 451 (c) 361 (d) 401
  25. 25.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
  26. 26.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
  27. 27.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3491 (c) 3481 (d) 3721
  28. 28.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 337 (c) 113 (d) 169
  29. 29.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 2 (c) 100 (d) 4
  30. 30.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 3 (c) 9 (d) 361
  31. 31.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 146 (c) 256 (d) 201
  32. 32.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 48 (d) 13
  33. 33.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 145 (c) 193 (d) 157
  34. 34.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x − 2)(x − 9)
  35. 35.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 15 (c) 13 (d) 14
  36. 36.If a + b = 10 and ab = 21, what is (a − b)²? (a) 16 (b) 58 (c) 4 (d) 100
  37. 37.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
  38. 38.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 29 (d) 31
  39. 39.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 1 and 21 (c) 1 and 9 (d) 2 and 8
  40. 40.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 145 (c) 217 (d) 289

Answer key

  1. 1521
  2. (x + 7)(x + 8)
  3. 4
  4. 48
  5. 25
  6. 9
  7. (x + 3)(x + 5)
  8. 13
  9. 6241
  10. 9
  11. (x + 8)(x + 9)
  12. 324
  13. 145
  14. 4096
  15. 50
  16. 1
  17. (x + 3)(x + 8)
  18. 4
  19. 181
  20. 15
  21. 10609
  22. 6889
  23. 74
  24. 441
  25. (x + 6)(x + 7)
  26. (x + 2)(x + 3)
  27. 3481
  28. 113
  29. 4
  30. 9
  31. 146
  32. 14
  33. 145
  34. (x + 2)(x + 9)
  35. 14
  36. 16
  37. (x + 6)(x + 9)
  38. 30
  39. 3 and 7
  40. 1

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