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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 781 · 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.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 15 (c) 8 (d) 9
  2. 2.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 5 (d) 4
  3. 3.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 63 (d) 16
  4. 4.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 3 (c) 25 (d) 17
  5. 5.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 289 (c) 7 (d) 169
  6. 6.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 225 (c) 137 (d) 181
  7. 7.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2401 (c) 2601 (d) 2611
  9. 9.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 441 (d) 451
  10. 10.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 875 (c) 625 (d) 1225
  11. 11.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  12. 12.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
  13. 13.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 149 (d) 9
  14. 14.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 21 (c) 10 (d) 9
  15. 15.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 896 (d) 784
  16. 16.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  17. 17.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 9996 (c) 9604 (d) 10404
  18. 18.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 72 (d) 73
  19. 19.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 3136 (c) 4096 (d) 4106
  20. 20.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 15 (d) 14
  21. 21.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 24 (c) 9 (d) 10
  22. 22.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 9)
  23. 23.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 144 (c) 112 (d) 208
  24. 24.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 84 (c) 68 (d) 132
  25. 25.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 11 (c) 13 (d) 12
  26. 26.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2714 (d) 2504
  27. 27.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
  28. 28.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)
  29. 29.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 145 (d) 1
  30. 30.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3364 (c) 3374 (d) 3844
  31. 31.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 1 (d) 169
  32. 32.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 729 (c) 1099 (d) 909
  33. 33.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
  34. 34.What are the roots of x² − 9x + 18 = 0? (a) 1 and 8 (b) 2 and 9 (c) 3 and 6 (d) 1 and 18
  35. 35.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 225 (c) 333 (d) 657
  36. 36.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 4884 (d) 5476
  37. 37.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 253 (c) 85 (d) 169
  38. 38.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 42 (d) 1 and 12
  39. 39.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 11 (c) 12 (d) 30
  40. 40.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 541 (d) 271

Answer key

  1. 8
  2. 5
  3. 16
  4. 9
  5. 49
  6. 137
  7. (x + 4)(x + 8)
  8. 2601
  9. 441
  10. 625
  11. (x + 6)(x + 7)
  12. (x + 2)(x + 4)
  13. 9
  14. 10
  15. 784
  16. (x + 8)(x + 9)
  17. 9604
  18. 72
  19. 4096
  20. 15
  21. 10
  22. (x + 3)(x + 7)
  23. 80
  24. 68
  25. 12
  26. 2704
  27. 2 and 3
  28. (x + 6)(x + 9)
  29. 1
  30. 3364
  31. 1
  32. 1089
  33. (x + 4)(x + 5)
  34. 3 and 6
  35. 225
  36. 4356
  37. 85
  38. 6 and 7
  39. 11
  40. 181

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