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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 231 · 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 = 18 and ab = 77, what is a² + b²? (a) 324 (b) 247 (c) 170 (d) 478
  2. 2.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 45 (c) 81 (d) 63
  3. 3.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3364 (c) 3854 (d) 3844
  4. 4.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 4 (d) 400
  5. 5.If a + b = 11 and ab = 24, what is (a − b)²? (a) 5 (b) 25 (c) 121 (d) 73
  6. 6.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 8 (c) 7 (d) 12
  7. 7.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 151 (c) 106 (d) 286
  8. 8.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3481 (c) 3721 (d) 3731
  9. 9.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 11 (d) 9
  10. 10.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1531 (c) 1599 (d) 1521
  11. 11.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 15 (c) 9 (d) 14
  12. 12.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 40 (c) 88 (d) 64
  13. 13.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
  14. 14.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 421 (d) 223
  15. 15.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 10201 (c) 9999 (d) 9811
  16. 16.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 9604 (c) 10004 (d) 10404
  17. 17.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2916 (c) 2926 (d) 2116
  18. 18.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 56 (d) 16
  19. 19.What is the product of the roots of x² − 12x + 32 = 0? (a) 33 (b) 32 (c) 12 (d) 31
  20. 20.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 73 (c) 65 (d) 97
  21. 21.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
  22. 22.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)
  23. 23.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 196 (c) 376 (d) 256
  24. 24.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 3 (d) 9
  25. 25.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 494 (c) 324 (d) 404
  26. 26.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1156 (d) 1166
  27. 27.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8836 (c) 8084 (d) 7396
  28. 28.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1681 (d) 1601
  29. 29.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 256 (c) 382 (d) 130
  30. 30.If a + b = 9 and ab = 14, what is (a − b)²? (a) 25 (b) 53 (c) 81 (d) 5
  31. 31.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
  32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  33. 33.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 145 (d) 1
  34. 34.If a + b = 7 and ab = 10, what is (a − b)²? (a) 9 (b) 3 (c) 49 (d) 29
  35. 35.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
  36. 36.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  37. 37.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3969 (c) 3609 (d) 3979
  38. 38.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)
  39. 39.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 198 (c) 90 (d) 144
  40. 40.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 16 (c) 256 (d) 4

Answer key

  1. 170
  2. 45
  3. 3844
  4. 4
  5. 25
  6. 8
  7. 106
  8. 3721
  9. 10
  10. 1521
  11. 14
  12. 40
  13. (x + 6)(x + 9)
  14. 157
  15. 9801
  16. 10404
  17. 2916
  18. 15
  19. 32
  20. 65
  21. (x + 2)(x + 8)
  22. (x + 7)(x + 9)
  23. 136
  24. 9
  25. 484
  26. 1156
  27. 7396
  28. 1681
  29. 130
  30. 25
  31. (x + 3)(x + 4)
  32. 17
  33. 1
  34. 9
  35. (x + 3)(x + 8)
  36. (x + 4)(x + 8)
  37. 3969
  38. (x + 2)(x + 3)
  39. 90
  40. 16

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