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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 501 · 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 102²? (a) 10404 (b) 10414 (c) 9604 (d) 10004
  2. 2.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 43 (c) 41 (d) 13
  3. 3.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 127 (c) 85 (d) 1
  4. 4.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 63 (d) 16
  5. 5.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
  6. 6.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 256 (c) 136 (d) 16
  7. 7.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 41 (c) 61 (d) 81
  8. 8.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
  9. 9.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 61 (c) 91 (d) 1
  10. 10.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 2 and 8 (c) 1 and 9 (d) 1 and 16
  11. 11.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 130 (c) 193 (d) 256
  12. 12.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 71 (d) 17
  13. 13.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 21 (c) 20 (d) 10
  14. 14.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1591 (c) 1849 (d) 1369
  15. 15.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 12 (c) 2 and 20 (d) 1 and 40
  16. 16.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 6425 (d) 7225
  17. 17.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 8649 (c) 8091 (d) 7579
  18. 18.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
  19. 19.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 125 (c) 169 (d) 213
  20. 20.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 8 (c) 1 and 5 (d) 2 and 4
  21. 21.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 625 (c) 1225 (d) 925
  22. 22.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 35 (c) 11 (d) 12
  23. 23.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
  24. 24.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3136 (c) 4096 (d) 3584
  25. 25.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 400 (d) 304
  26. 26.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 61 (c) 37 (d) 43
  27. 27.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)
  28. 28.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  29. 29.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
  30. 30.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 9 (c) 13 (d) 7
  31. 31.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2916 (c) 2516 (d) 2926
  32. 32.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 11 (c) 13 (d) 12
  33. 33.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2509 (c) 2819 (d) 2209
  34. 34.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 85 (c) 169 (d) 253
  35. 35.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2916 (c) 2484 (d) 2126
  36. 36.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 14 (c) 15 (d) 8
  37. 37.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 676 (c) 916 (d) 1166
  38. 38.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 14 (c) 6 and 9 (d) 1 and 54
  39. 39.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 299 (d) 391
  40. 40.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 6 (c) 36 (d) 64

Answer key

  1. 10404
  2. 42
  3. 1
  4. 63
  5. (x + 4)(x + 6)
  6. 16
  7. 1
  8. (x + 6)(x + 9)
  9. 1
  10. 2 and 8
  11. 130
  12. 72
  13. 21
  14. 1369
  15. 5 and 8
  16. 7225
  17. 7569
  18. (x + 3)(x + 6)
  19. 125
  20. 2 and 4
  21. 1225
  22. 12
  23. (x + 4)(x + 9)
  24. 3136
  25. 208
  26. 37
  27. (x + 7)(x + 9)
  28. (x + 5)(x + 8)
  29. 5 and 6
  30. 5
  31. 2916
  32. 12
  33. 2809
  34. 85
  35. 2116
  36. 15
  37. 1156
  38. 6 and 9
  39. 289
  40. 36

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