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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 938 · 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.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 5 and 6 (c) 2 and 15 (d) 1 and 10
  2. 2.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 61 (c) 181 (d) 121
  3. 3.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10211 (c) 10201 (d) 9801
  4. 4.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
  5. 5.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 1 and 15 (c) 2 and 6 (d) 1 and 7
  6. 6.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 12 (d) 11
  7. 7.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  8. 8.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 7 (c) 169 (d) 49
  9. 9.What is the product of the roots of x² − 7x + 10 = 0? (a) 10 (b) 7 (c) 11 (d) 9
  10. 10.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 4 (c) 400 (d) 2
  11. 11.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 117 (c) 225 (d) 171
  12. 12.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 8 (c) 7 (d) 6
  13. 13.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 729 (d) 1089
  14. 14.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 164 (d) 244
  15. 15.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 400 (d) 592
  16. 16.If a + b = 8 and ab = 7, what is (a − b)²? (a) 6 (b) 36 (c) 50 (d) 64
  17. 17.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 576 (c) 586 (d) 256
  18. 18.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 181 (d) 1
  19. 19.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2219 (c) 2491 (d) 2209
  20. 20.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 334 (c) 396 (d) 324
  21. 21.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
  22. 22.What are the roots of x² − 15x + 54 = 0? (a) 1 and 14 (b) 6 and 9 (c) 1 and 54 (d) 2 and 27
  23. 23.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 635 (c) 625 (d) 875
  24. 24.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 299 (c) 391 (d) 529
  25. 25.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8116 (d) 8836
  26. 26.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
  27. 27.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 63 (c) 45 (d) 81
  28. 28.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 64 (c) 52 (d) 40
  29. 29.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)
  30. 30.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4875 (c) 5625 (d) 4235
  31. 31.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 5)(x + 7) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
  32. 32.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 1 and 63 (d) 7 and 9
  33. 33.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 324 (c) 170 (d) 478
  34. 34.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 961 (d) 841
  35. 35.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 3 (d) 441
  36. 36.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 37 (c) 61 (d) 43
  37. 37.If a + b = 10 and ab = 9, what is a² + b²? (a) 118 (b) 100 (c) 82 (d) 91
  38. 38.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 1 and 11 (c) 1 and 35 (d) 2 and 10
  39. 39.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
  40. 40.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 25 (c) 5 (d) 225

Answer key

  1. 5 and 6
  2. 61
  3. 10201
  4. (x + 5)(x + 9)
  5. 3 and 5
  6. 11
  7. (x + 7)(x + 9)
  8. 49
  9. 10
  10. 4
  11. 117
  12. 6
  13. 1089
  14. 164
  15. 208
  16. 36
  17. 576
  18. 1
  19. 2209
  20. 324
  21. (x + 7)(x + 8)
  22. 6 and 9
  23. 625
  24. 289
  25. 8836
  26. (x + 6)(x + 9)
  27. 45
  28. 40
  29. (x + 3)(x + 6)
  30. 4225
  31. (x + 5)(x + 7)
  32. 7 and 9
  33. 170
  34. 961
  35. 9
  36. 37
  37. 82
  38. 5 and 7
  39. (x + 6)(x + 8)
  40. 25

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