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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 10 · 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 61²? (a) 3731 (b) 3721 (c) 3481 (d) 3601
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  3. 3.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 3025 (d) 2475
  4. 4.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1531 (c) 1599 (d) 1681
  5. 5.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 468 (c) 324 (d) 180
  6. 6.If a + b = 8 and ab = 7, what is a² + b²? (a) 57 (b) 78 (c) 64 (d) 50
  7. 7.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 6 (c) 36 (d) 256
  8. 8.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2314 (c) 2496 (d) 2704
  9. 9.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 396 (c) 324 (d) 334
  10. 10.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 896 (d) 1024
  11. 11.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 286 (c) 151 (d) 196
  12. 12.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 7 (c) 5 (d) 9
  13. 13.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 7569 (c) 8649 (d) 8109
  14. 14.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 9604 (c) 10404 (d) 10004
  15. 15.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  16. 16.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 71 (d) 17
  17. 17.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 12 (c) 11 (d) 30
  18. 18.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)
  19. 19.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)
  20. 20.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  21. 21.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 225 (d) 657
  22. 22.If a + b = 12 and ab = 20, what is a² + b²? (a) 104 (b) 184 (c) 144 (d) 124
  23. 23.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5041 (c) 5051 (d) 4901
  24. 24.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2491 (c) 2209 (d) 2809
  25. 25.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 48 (d) 14
  26. 26.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 10) (c) (x + 1)(x + 6) (d) (x + 2)(x + 5)
  27. 27.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 20 (c) 10 (d) 9
  28. 28.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 10404 (c) 9996 (d) 9614
  29. 29.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 223 (c) 421 (d) 289
  30. 30.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 441 (c) 401 (d) 361
  31. 31.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 5 and 7 (c) 1 and 11 (d) 1 and 35
  32. 32.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
  33. 33.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 3 (c) 225 (d) 9
  34. 34.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 271 (d) 361
  35. 35.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 11 (c) 12 (d) 24
  36. 36.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 4 (c) 5 (d) 6
  37. 37.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1024 (c) 1034 (d) 904
  38. 38.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)
  39. 39.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 29 (c) 30 (d) 11
  40. 40.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 193 (c) 382 (d) 130

Answer key

  1. 3721
  2. (x + 8)(x + 9)
  3. 2025
  4. 1521
  5. 180
  6. 50
  7. 36
  8. 2304
  9. 324
  10. 784
  11. 106
  12. 1
  13. 8649
  14. 10404
  15. (x + 6)(x + 7)
  16. 72
  17. 11
  18. (x + 5)(x + 6)
  19. (x + 3)(x + 5)
  20. (x + 7)(x + 8)
  21. 225
  22. 104
  23. 5041
  24. 2209
  25. 48
  26. (x + 2)(x + 5)
  27. 9
  28. 9604
  29. 157
  30. 441
  31. 5 and 7
  32. (x + 4)(x + 9)
  33. 9
  34. 1
  35. 11
  36. 5
  37. 1024
  38. (x + 5)(x + 9)
  39. 30
  40. 130

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