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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 558 · 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 + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
  2. 2.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
  3. 3.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 137 (c) 313 (d) 181
  4. 4.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 9 (d) 149
  5. 5.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 441 (c) 361 (d) 401
  6. 6.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 198 (c) 144 (d) 90
  7. 7.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 4096 (c) 3136 (d) 3616
  9. 9.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 100 (c) 79 (d) 58
  10. 10.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 8 (c) 1 and 20 (d) 4 and 5
  11. 11.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
  12. 12.If a + b = 10 and ab = 16, what is (a − b)²? (a) 36 (b) 100 (c) 6 (d) 68
  13. 13.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 3 and 8 (d) 1 and 10
  14. 14.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 304 (c) 208 (d) 592
  15. 15.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 5 (c) 1 (d) 9
  16. 16.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 425 (c) 625 (d) 635
  17. 17.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 635 (c) 875 (d) 625
  18. 18.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 3 (c) 361 (d) 185
  19. 19.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 5 (c) 289 (d) 25
  20. 20.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 8 (c) 10 (d) 6
  21. 21.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4225 (c) 3025 (d) 4235
  22. 22.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 3 and 4 (c) 2 and 6 (d) 1 and 6
  23. 23.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 253 (c) 85 (d) 169
  24. 24.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 63 (d) 15
  25. 25.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x − 2)(x − 8)
  26. 26.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 4 (c) 256 (d) 16
  27. 27.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)
  28. 28.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 169 (c) 289 (d) 409
  29. 29.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 9 (c) 11 (d) 16
  30. 30.What is the product of the roots of x² − 12x + 35 = 0? (a) 35 (b) 34 (c) 36 (d) 12
  31. 31.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9409 (b) 9419 (c) 9991 (d) 10609
  32. 32.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 121 (c) 73 (d) 169
  33. 33.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
  34. 34.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x − 3)(x − 5)
  35. 35.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 11 (d) 13
  36. 36.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4356 (c) 4366 (d) 4884
  37. 37.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 15 (d) 56
  38. 38.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 106 (c) 16 (d) 4
  39. 39.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
  40. 40.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 2 (c) 324 (d) 4

Answer key

  1. (x + 4)(x + 7)
  2. (x + 5)(x + 8)
  3. 137
  4. 9
  5. 441
  6. 90
  7. (x + 7)(x + 8)
  8. 4096
  9. 58
  10. 4 and 5
  11. (x + 6)(x + 9)
  12. 36
  13. 3 and 8
  14. 208
  15. 1
  16. 625
  17. 625
  18. 9
  19. 25
  20. 7
  21. 4225
  22. 3 and 4
  23. 85
  24. 16
  25. (x + 2)(x + 8)
  26. 16
  27. (x + 8)(x + 9)
  28. 169
  29. 10
  30. 35
  31. 9409
  32. 73
  33. (x + 3)(x + 6)
  34. (x + 3)(x + 5)
  35. 12
  36. 4356
  37. 15
  38. 16
  39. (x + 2)(x + 5)
  40. 4

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