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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 976 · 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 + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2601 (c) 2401 (d) 2499
  3. 3.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
  4. 4.What is the sum of the roots of x² − 8x + 12 = 0? (a) 8 (b) 7 (c) 12 (d) 9
  5. 5.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
  6. 6.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)
  7. 7.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 89 (c) 129 (d) 169
  8. 8.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2401 (c) 2601 (d) 2611
  9. 9.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x + 1)(x + 12)
  10. 10.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 1 and 12 (c) 5 and 8 (d) 2 and 20
  11. 11.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1936 (c) 1306 (d) 1296
  13. 13.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 400 (d) 598
  14. 14.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 202 (d) 400
  15. 15.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 184 (c) 104 (d) 124
  16. 16.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1531 (c) 1681 (d) 1521
  17. 17.If a + b = 11 and ab = 18, what is a² + b²? (a) 85 (b) 103 (c) 157 (d) 121
  18. 18.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576
  19. 19.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 11 (c) 10 (d) 7
  20. 20.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 64 (c) 34 (d) 94
  21. 21.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
  22. 22.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 196 (c) 376 (d) 256
  23. 23.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 16 (d) 1 and 72
  24. 24.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 241 (c) 133 (d) 97
  25. 25.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 225 (c) 441 (d) 3
  26. 26.If a + b = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 8 (c) 82 (d) 64
  27. 27.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 9 (d) 10
  28. 28.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 1 and 10 (c) 2 and 15 (d) 5 and 6
  29. 29.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 529 (c) 361 (d) 277
  30. 30.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 21 (c) 20 (d) 10
  31. 31.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
  32. 32.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
  33. 33.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)
  34. 34.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 47 (c) 14 (d) 49
  35. 35.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2926 (c) 2116 (d) 2516
  36. 36.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 11 (c) 4 and 8 (d) 1 and 32
  37. 37.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 14 (d) 16
  38. 38.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 2 (c) 74 (d) 4
  39. 39.If a + b = 9 and ab = 8, what is (a − b)²? (a) 7 (b) 65 (c) 49 (d) 81
  40. 40.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 16 (c) 36 (d) 4

Answer key

  1. (x + 6)(x + 8)
  2. 2401
  3. (x + 5)(x + 6)
  4. 8
  5. (x + 5)(x + 8)
  6. (x + 2)(x + 4)
  7. 89
  8. 2601
  9. (x + 4)(x + 9)
  10. 5 and 8
  11. (x + 7)(x + 9)
  12. 1296
  13. 202
  14. 4
  15. 104
  16. 1521
  17. 85
  18. 576
  19. 10
  20. 34
  21. (x + 5)(x + 9)
  22. 136
  23. 8 and 9
  24. 97
  25. 9
  26. 64
  27. 10
  28. 5 and 6
  29. 193
  30. 21
  31. (x + 3)(x + 7)
  32. (x + 4)(x + 6)
  33. (x + 8)(x + 9)
  34. 48
  35. 2916
  36. 4 and 8
  37. 15
  38. 4
  39. 49
  40. 16

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