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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 388 · 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 = 15 and ab = 36, what is a² + b²? (a) 153 (b) 297 (c) 189 (d) 225
  2. 2.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 2 and 4 (c) 2 and 3 (d) 1 and 6
  3. 3.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
  4. 4.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 208 (d) 304
  5. 5.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 44 (c) 14 (d) 45
  6. 6.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  7. 7.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 61 (c) 121 (d) 41
  8. 8.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 121 (c) 141 (d) 101
  9. 9.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
  10. 10.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 2 and 8 (c) 1 and 21 (d) 3 and 7
  11. 11.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 14 (c) 40 (d) 13
  12. 12.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 225 (c) 113 (d) 169
  13. 13.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x − 2)(x − 8)
  14. 14.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 54 (c) 15 (d) 16
  15. 15.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
  16. 16.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 62 (d) 63
  17. 17.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 3 (d) 9
  18. 18.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 1 and 30 (c) 5 and 6 (d) 2 and 15
  19. 19.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 130 (c) 4 (d) 256
  20. 20.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 10 (c) 6 (d) 8
  21. 21.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 9 (c) 7 (d) 13
  22. 22.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 10816 (c) 9984 (d) 9216
  23. 23.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9419 (c) 9991 (d) 9409
  24. 24.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 11 (c) 27 (d) 28
  25. 25.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  26. 26.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 17 (d) 63
  27. 27.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  28. 28.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 324 (c) 4 (d) 170
  29. 29.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 1034 (d) 784
  30. 30.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4366 (c) 5476 (d) 4356
  31. 31.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 91 (c) 121 (d) 61
  32. 32.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6396 (c) 6724 (d) 6094
  33. 33.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9801 (c) 9999 (d) 10201
  34. 34.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2704 (c) 2304 (d) 2504
  35. 35.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 2 and 10 (c) 4 and 5 (d) 1 and 20
  36. 36.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8464 (c) 8096 (d) 7754
  37. 37.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 19 (c) 25 (d) 13
  38. 38.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 1 and 11 (c) 2 and 10 (d) 1 and 35
  39. 39.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3374 (c) 3844 (d) 3364
  40. 40.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)

Answer key

  1. 153
  2. 2 and 3
  3. (x + 5)(x + 9)
  4. 208
  5. 45
  6. (x + 6)(x + 7)
  7. 41
  8. 101
  9. (x + 6)(x + 9)
  10. 3 and 7
  11. 13
  12. 113
  13. (x + 2)(x + 8)
  14. 15
  15. (x + 3)(x + 6)
  16. 63
  17. 9
  18. 5 and 6
  19. 4
  20. 7
  21. 5
  22. 9216
  23. 9409
  24. 28
  25. 17
  26. 16
  27. (x + 8)(x + 9)
  28. 16
  29. 1024
  30. 4356
  31. 61
  32. 6084
  33. 9801
  34. 2704
  35. 4 and 5
  36. 7744
  37. 1
  38. 5 and 7
  39. 3364
  40. (x + 7)(x + 9)

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