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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 64 · 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 = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 9 (c) 89 (d) 3
  2. 2.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6724 (c) 6734 (d) 6084
  4. 4.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
  5. 5.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 13 (c) 14 (d) 36
  6. 6.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 14 (d) 16
  7. 7.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 4 (c) 64 (d) 34
  8. 8.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 971 (d) 961
  9. 9.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
  10. 10.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 271 (d) 361
  11. 11.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 169 (c) 129 (d) 89
  12. 12.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 5 (c) 25 (d) 169
  13. 13.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 1 and 63 (d) 3 and 21
  14. 14.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 925 (c) 1225 (d) 1235
  15. 15.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 576 (c) 586 (d) 416
  16. 16.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 4 and 8 (c) 1 and 11 (d) 1 and 32
  17. 17.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 2)(x + 7) (c) (x − 2)(x − 7) (d) (x + 1)(x + 14)
  18. 18.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 271 (c) 361 (d) 181
  19. 19.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9419 (c) 9409 (d) 10609
  20. 20.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 49 (c) 69 (d) 29
  21. 21.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
  22. 22.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 73 (c) 65 (d) 81
  23. 23.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 7 (c) 6 (d) 4
  24. 24.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 27 (c) 2 and 10 (d) 1 and 11
  25. 25.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2501 (c) 2611 (d) 2401
  26. 26.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 11 (c) 10 (d) 24
  27. 27.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 225 (c) 137 (d) 181
  28. 28.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 8 (c) 64 (d) 104
  29. 29.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 875 (c) 635 (d) 625
  30. 30.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 113 (d) 337
  31. 31.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  32. 32.What is the sum of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 8 (d) 6
  33. 33.If a + b = 11 and ab = 18, what is a² + b²? (a) 85 (b) 121 (c) 157 (d) 103
  34. 34.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2601 (c) 2401 (d) 2411
  35. 35.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
  36. 36.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 97 (c) 73 (d) 169
  37. 37.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  38. 38.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 4 and 6 (c) 1 and 24 (d) 2 and 12
  39. 39.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 676 (d) 1156
  40. 40.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 10 (d) 12

Answer key

  1. 9
  2. (x + 6)(x + 7)
  3. 6724
  4. (x + 4)(x + 8)
  5. 13
  6. 15
  7. 4
  8. 961
  9. (x + 5)(x + 6)
  10. 181
  11. 89
  12. 25
  13. 7 and 9
  14. 1225
  15. 576
  16. 4 and 8
  17. (x + 2)(x + 7)
  18. 1
  19. 9409
  20. 29
  21. (x + 3)(x + 8)
  22. 65
  23. 5
  24. 3 and 9
  25. 2601
  26. 11
  27. 137
  28. 64
  29. 625
  30. 113
  31. (x + 8)(x + 9)
  32. 7
  33. 85
  34. 2401
  35. 16
  36. 73
  37. (x + 7)(x + 8)
  38. 4 and 6
  39. 676
  40. 11

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