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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 524 · 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 43²? (a) 1849 (b) 1609 (c) 1369 (d) 1859
  2. 2.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 361 (d) 181
  3. 3.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 15 (d) 55
  4. 4.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 117 (c) 198 (d) 90
  5. 5.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1604 (c) 1444 (d) 1774
  6. 6.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)
  7. 7.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
  8. 8.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 4 and 7 (c) 1 and 28 (d) 1 and 10
  9. 9.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 1)(x + 10)
  10. 10.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
  11. 11.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 7 (c) 8 (d) 12
  12. 12.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)
  13. 13.If a + b = 12 and ab = 20, what is a² + b²? (a) 104 (b) 184 (c) 144 (d) 124
  14. 14.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
  15. 15.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
  16. 16.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 7235 (c) 9025 (d) 8075
  17. 17.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 7 and 8 (d) 1 and 14
  18. 18.If a + b = 16 and ab = 60, what is (a − b)²? (a) 136 (b) 256 (c) 16 (d) 4
  19. 19.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  20. 20.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 1 and 14 (c) 1 and 8 (d) 3 and 6
  21. 21.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 76 (c) 100 (d) 52
  22. 22.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 25 (c) 3 (d) 17
  23. 23.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1369 (c) 1849 (d) 1379
  24. 24.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 169 (c) 125 (d) 213
  25. 25.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 901 (d) 961
  26. 26.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 15 (c) 45 (d) 14
  27. 27.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 676 (d) 1156
  28. 28.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 169 (c) 127 (d) 253
  29. 29.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 4 (c) 16 (d) 36
  30. 30.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 8 (c) 64 (d) 144
  31. 31.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 56 (d) 16
  32. 32.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3979 (c) 3969 (d) 3249
  33. 33.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 225 (d) 441
  34. 34.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7225 (c) 5625 (d) 7235
  35. 35.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8084 (c) 7406 (d) 8836
  36. 36.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 1 and 7 (c) 3 and 5 (d) 2 and 6
  37. 37.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 153 (c) 81 (d) 9
  38. 38.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2491 (c) 2209 (d) 2809
  39. 39.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
  40. 40.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)

Answer key

  1. 1849
  2. 181
  3. 56
  4. 90
  5. 1764
  6. (x + 7)(x + 9)
  7. (x + 3)(x + 7)
  8. 4 and 7
  9. (x + 3)(x + 8)
  10. (x + 4)(x + 5)
  11. 7
  12. (x + 8)(x + 9)
  13. 104
  14. (x + 2)(x + 3)
  15. (x + 3)(x + 5)
  16. 7225
  17. 7 and 8
  18. 16
  19. (x + 5)(x + 7)
  20. 2 and 7
  21. 52
  22. 9
  23. 1369
  24. 125
  25. 961
  26. 14
  27. 676
  28. 85
  29. 16
  30. 64
  31. 15
  32. 3969
  33. 225
  34. 7225
  35. 7396
  36. 3 and 5
  37. 81
  38. 2209
  39. (x + 2)(x + 6)
  40. (x + 4)(x + 6)

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