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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 340 · 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 + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
  2. 2.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)
  3. 3.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 9 (c) 16 (d) 10
  4. 4.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  5. 5.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 48 (c) 13 (d) 15
  6. 6.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
  7. 7.If a + b = 10 and ab = 24, what is a² + b²? (a) 52 (b) 76 (c) 100 (d) 148
  8. 8.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
  9. 9.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6571 (d) 6241
  10. 10.If a + b = 10 and ab = 9, what is a² + b²? (a) 91 (b) 82 (c) 100 (d) 118
  11. 11.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2116 (c) 2126 (d) 2916
  12. 12.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
  13. 13.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)
  14. 14.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 73 (d) 72
  15. 15.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 47 (c) 14 (d) 49
  16. 16.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 72 (d) 1 and 16
  17. 17.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
  18. 18.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 6 (c) 7 (d) 10
  19. 19.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 10 (c) 3 and 8 (d) 1 and 24
  20. 20.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 7569 (c) 8659 (d) 8109
  21. 21.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 25 (c) 5 (d) 361
  22. 22.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 104 (c) 64 (d) 8
  23. 23.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
  24. 24.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2496 (c) 2304 (d) 2314
  25. 25.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 586 (d) 576
  26. 26.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6251 (c) 6399 (d) 6241
  27. 27.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 26 (c) 36 (d) 46
  28. 28.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
  29. 29.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 3 (c) 361 (d) 9
  30. 30.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 676 (c) 1166 (d) 916
  31. 31.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
  32. 32.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 961 (c) 971 (d) 841
  33. 33.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 6416 (c) 7056 (d) 5776
  34. 34.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 12 (d) 11
  35. 35.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 7 (c) 289 (d) 49
  36. 36.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 53 (d) 54
  37. 37.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
  38. 38.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 9409 (c) 10009 (d) 10609
  39. 39.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 2 (c) 130 (d) 4
  40. 40.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 20 (c) 10 (d) 21

Answer key

  1. (x + 3)(x + 9)
  2. (x + 4)(x + 8)
  3. 10
  4. (x + 5)(x + 8)
  5. 14
  6. (x + 2)(x + 9)
  7. 52
  8. 441
  9. 6561
  10. 82
  11. 2116
  12. (x + 4)(x + 5)
  13. (x + 7)(x + 8)
  14. 72
  15. 48
  16. 8 and 9
  17. (x + 4)(x + 9)
  18. 7
  19. 3 and 8
  20. 8649
  21. 25
  22. 64
  23. (x + 2)(x + 7)
  24. 2304
  25. 576
  26. 6241
  27. 26
  28. (x + 2)(x + 6)
  29. 9
  30. 1156
  31. (x + 2)(x + 4)
  32. 961
  33. 7056
  34. 11
  35. 49
  36. 54
  37. (x + 2)(x + 8)
  38. 10609
  39. 4
  40. 21

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