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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 680 · 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 = 11 and ab = 28, what is (a − b)²? (a) 9 (b) 121 (c) 65 (d) 3
  2. 2.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2304 (d) 2314
  3. 3.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 6425 (d) 5625
  4. 4.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 576 (d) 586
  5. 5.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 29 (c) 69 (d) 49
  6. 6.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
  7. 7.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 14 (d) 16
  8. 8.If a + b = 13 and ab = 12, what is (a − b)²? (a) 169 (b) 121 (c) 145 (d) 11
  9. 9.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  10. 10.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 136 (c) 4 (d) 256
  11. 11.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 14 (c) 13 (d) 42
  12. 12.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 130 (c) 64 (d) 8
  13. 13.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 289 (c) 3 (d) 9
  14. 14.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 4 (d) 2
  15. 15.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 337 (c) 225 (d) 113
  16. 16.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 289 (c) 169 (d) 49
  17. 17.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 1 (d) 113
  18. 18.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 3 (c) 81 (d) 9
  19. 19.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 148 (c) 172 (d) 196
  20. 20.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)
  21. 21.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
  22. 22.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
  23. 23.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 4 (d) 16
  24. 24.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 256 (c) 130 (d) 2
  25. 25.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 8281 (d) 7921
  26. 26.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 41 (c) 39 (d) 13
  27. 27.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 1034 (d) 784
  28. 28.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 11 (c) 13 (d) 12
  29. 29.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 9 (c) 10 (d) 11
  30. 30.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 72 (d) 71
  31. 31.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
  32. 32.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 81 (c) 67 (d) 109
  33. 33.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 333 (c) 225 (d) 441
  34. 34.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 5 (c) 7 (d) 8
  35. 35.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 9 (c) 8 (d) 15
  36. 36.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 10404 (c) 9604 (d) 9614
  38. 38.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
  39. 39.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 1)(x + 6) (d) (x + 2)(x + 5)
  40. 40.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3491 (c) 3599 (d) 3481

Answer key

  1. 9
  2. 2304
  3. 7225
  4. 576
  5. 29
  6. (x + 2)(x + 4)
  7. 15
  8. 121
  9. (x + 6)(x + 7)
  10. 16
  11. 13
  12. 64
  13. 9
  14. 4
  15. 113
  16. 49
  17. 1
  18. 9
  19. 148
  20. (x + 5)(x + 8)
  21. (x + 6)(x + 9)
  22. (x + 3)(x + 9)
  23. 16
  24. 4
  25. 8281
  26. 40
  27. 1024
  28. 12
  29. 10
  30. 72
  31. (x + 2)(x + 6)
  32. 53
  33. 225
  34. 6
  35. 8
  36. (x + 8)(x + 9)
  37. 9604
  38. (x + 4)(x + 7)
  39. (x + 2)(x + 5)
  40. 3481

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