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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 683 · 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.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 5329 (c) 4909 (d) 4489
  2. 2.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3035 (c) 3025 (d) 2025
  3. 3.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 9 (c) 7 (d) 15
  4. 4.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 52 (c) 40 (d) 88
  5. 5.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 16 (d) 14
  6. 6.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 15 (d) 55
  7. 7.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2714 (c) 2304 (d) 2504
  8. 8.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
  9. 9.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 145 (c) 169 (d) 121
  10. 10.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 400 (c) 208 (d) 592
  11. 11.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 271 (c) 1 (d) 181
  12. 12.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 100 (c) 2 (d) 4
  13. 13.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)
  14. 14.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 13 (d) 14
  15. 15.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 8 (d) 10
  16. 16.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
  17. 17.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 401 (c) 361 (d) 441
  18. 18.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 145 (c) 289 (d) 217
  19. 19.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 6 (c) 324 (d) 36
  20. 20.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 1 and 8 (c) 2 and 10 (d) 1 and 20
  21. 21.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
  22. 22.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  23. 23.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 43 (c) 42 (d) 41
  24. 24.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
  25. 25.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 36 (c) 35 (d) 13
  26. 26.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 10404 (c) 9604 (d) 9996
  27. 27.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 333 (c) 225 (d) 171
  28. 28.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  29. 29.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
  30. 30.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 20 (c) 21 (d) 22
  31. 31.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
  32. 32.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
  33. 33.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 5476 (c) 4366 (d) 4356
  34. 34.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 8 (c) 160 (d) 256
  35. 35.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 1089 (d) 729
  36. 36.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 484 (d) 494
  37. 37.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 62 (d) 64
  38. 38.If a + b = 17 and ab = 60, what is (a − b)²? (a) 7 (b) 49 (c) 289 (d) 169
  39. 39.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 361 (c) 193 (d) 529
  40. 40.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)

Answer key

  1. 5329
  2. 3025
  3. 8
  4. 40
  5. 15
  6. 56
  7. 2704
  8. (x + 4)(x + 8)
  9. 121
  10. 208
  11. 1
  12. 4
  13. (x + 7)(x + 9)
  14. 13
  15. 9
  16. 16
  17. 441
  18. 1
  19. 36
  20. 4 and 5
  21. (x + 2)(x + 6)
  22. (x + 5)(x + 7)
  23. 42
  24. 17
  25. 36
  26. 9604
  27. 117
  28. (x + 7)(x + 8)
  29. (x + 2)(x + 4)
  30. 21
  31. (x + 5)(x + 8)
  32. (x + 3)(x + 6)
  33. 4356
  34. 64
  35. 729
  36. 484
  37. 63
  38. 49
  39. 193
  40. (x + 5)(x + 6)

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