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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 856 · 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 = 10 and ab = 16, what is a² + b²? (a) 100 (b) 68 (c) 84 (d) 132
  2. 2.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 7225 (d) 5625
  3. 3.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 181 (c) 121 (d) 91
  4. 4.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 425 (c) 625 (d) 635
  5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
  6. 6.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2314 (d) 2304
  7. 7.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 74 (c) 144 (d) 2
  8. 8.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6251 (d) 6241
  9. 9.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 72 (d) 16
  10. 10.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 13 (d) 14
  11. 11.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 249 (c) 89 (d) 129
  12. 12.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  13. 13.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 217 (d) 145
  14. 14.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1849 (c) 1609 (d) 1369
  15. 15.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 14 (c) 48 (d) 15
  16. 16.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
  17. 17.What are the roots of x² − 12x + 32 = 0? (a) 1 and 32 (b) 2 and 16 (c) 4 and 8 (d) 1 and 11
  18. 18.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
  19. 19.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 116 (c) 196 (d) 276
  20. 20.If a + b = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 10 (c) 100 (d) 148
  21. 21.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 144 (c) 117 (d) 198
  22. 22.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3136 (c) 3146 (d) 3584
  23. 23.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 25 (c) 125 (d) 225
  24. 24.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2516 (c) 2926 (d) 2916
  25. 25.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 35 (c) 13 (d) 12
  26. 26.If a + b = 13 and ab = 22, what is (a − b)²? (a) 169 (b) 81 (c) 125 (d) 9
  27. 27.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 5 (c) 25 (d) 157
  28. 28.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 14 (c) 15 (d) 16
  29. 29.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 9 (c) 5 (d) 1
  30. 30.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
  31. 31.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 2 and 27 (c) 6 and 9 (d) 1 and 14
  32. 32.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 256 (d) 130
  33. 33.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 17 (d) 15
  34. 34.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3481 (b) 3599 (c) 3721 (d) 3491
  35. 35.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 8 and 9 (d) 1 and 16
  36. 36.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 48 (c) 1 and 13 (d) 6 and 8
  37. 37.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9975 (c) 11025 (d) 9025
  38. 38.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 5 and 8 (c) 1 and 40 (d) 2 and 20
  39. 39.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 598 (d) 202
  40. 40.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 6384 (c) 5776 (d) 7056

Answer key

  1. 68
  2. 7225
  3. 61
  4. 625
  5. (x + 7)(x + 9)
  6. 2304
  7. 4
  8. 6241
  9. 17
  10. 13
  11. 89
  12. (x + 8)(x + 9)
  13. 1
  14. 1849
  15. 14
  16. (x + 3)(x + 6)
  17. 4 and 8
  18. (x + 3)(x + 9)
  19. 116
  20. 100
  21. 90
  22. 3136
  23. 25
  24. 2916
  25. 12
  26. 81
  27. 25
  28. 15
  29. 1
  30. (x + 5)(x + 6)
  31. 6 and 9
  32. 130
  33. 16
  34. 3481
  35. 8 and 9
  36. 6 and 8
  37. 9025
  38. 5 and 8
  39. 202
  40. 5776

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