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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 972 · 40 questions
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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 91²? (a) 8291 (b) 8281 (c) 8101 (d) 7921
  2. 2.If a + b = 13 and ab = 12, what is (a − b)²? (a) 169 (b) 121 (c) 145 (d) 11
  3. 3.What is the sum of the roots of x² − 8x + 15 = 0? (a) 15 (b) 7 (c) 8 (d) 9
  4. 4.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 371 (c) 361 (d) 441
  5. 5.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4899 (c) 5041 (d) 4771
  6. 6.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
  7. 7.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 361 (d) 193
  8. 8.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 635 (c) 625 (d) 225
  9. 9.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 157 (c) 25 (d) 289
  10. 10.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 18) (c) (x + 1)(x + 10) (d) (x + 2)(x + 9)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3596 (c) 3374 (d) 3844
  12. 12.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 13 (c) 15 (d) 9
  13. 13.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 144 (c) 133 (d) 122
  14. 14.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 1 and 35 (c) 5 and 7 (d) 2 and 10
  15. 15.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x − 4)(x − 9) (c) (x + 1)(x + 12) (d) (x + 1)(x + 36)
  16. 16.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 5339 (c) 4909 (d) 4489
  17. 17.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 10201 (c) 9801 (d) 9999
  19. 19.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  20. 20.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1616 (d) 1936
  21. 21.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
  22. 22.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 71 (d) 17
  23. 23.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 196 (c) 148 (d) 10
  24. 24.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 217 (c) 289 (d) 145
  25. 25.What is the product of the roots of x² − 10x + 16 = 0? (a) 15 (b) 10 (c) 16 (d) 17
  26. 26.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
  27. 27.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7225 (c) 7235 (d) 6425
  28. 28.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 401 (c) 361 (d) 441
  29. 29.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 117 (c) 225 (d) 9
  30. 30.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 28 (c) 20 (d) 52
  31. 31.If a + b = 12 and ab = 11, what is (a − b)²? (a) 100 (b) 144 (c) 122 (d) 10
  32. 32.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4896 (c) 4634 (d) 4624
  33. 33.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 2 and 24 (c) 6 and 8 (d) 1 and 48
  34. 34.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 16 (c) 10 (d) 11
  35. 35.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 10 (c) 1 and 6 (d) 2 and 5
  36. 36.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 4225 (c) 3025 (d) 3035
  38. 38.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 324 (c) 36 (d) 180
  39. 39.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4235 (c) 3025 (d) 4225
  40. 40.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 635 (c) 875 (d) 625

Answer key

  1. 8281
  2. 121
  3. 8
  4. 361
  5. 4761
  6. (x + 2)(x + 3)
  7. 193
  8. 625
  9. 25
  10. (x + 2)(x + 9)
  11. 3364
  12. 14
  13. 122
  14. 5 and 7
  15. (x + 4)(x + 9)
  16. 5329
  17. (x + 2)(x + 8)
  18. 9801
  19. (x + 4)(x + 8)
  20. 1936
  21. 8 and 9
  22. 72
  23. 100
  24. 1
  25. 16
  26. (x + 6)(x + 9)
  27. 7225
  28. 441
  29. 9
  30. 20
  31. 100
  32. 4624
  33. 6 and 8
  34. 10
  35. 2 and 5
  36. (x + 3)(x + 8)
  37. 3025
  38. 36
  39. 4225
  40. 625

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