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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 901 · 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 = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 169 (c) 121 (d) 11
  2. 2.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 20 (c) 52 (d) 28
  3. 3.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 391 (c) 289 (d) 529
  4. 4.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 49 (c) 73 (d) 37
  5. 5.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 85 (c) 7 (d) 49
  6. 6.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 4 and 5 (c) 1 and 8 (d) 2 and 10
  7. 7.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
  8. 8.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181
  9. 9.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 144 (c) 74 (d) 214
  10. 10.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 10201 (c) 9999 (d) 9801
  11. 11.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 9 (c) 169 (d) 89
  12. 12.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 409 (d) 529
  13. 13.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 9409 (c) 10009 (d) 10609
  14. 14.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 16 (c) 14 (d) 56
  15. 15.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
  16. 16.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3259 (c) 3249 (d) 3591
  17. 17.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
  18. 18.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 7921 (d) 8281
  19. 19.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 441 (d) 451
  20. 20.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
  21. 21.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 14 (c) 8 (d) 15
  22. 22.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 41 (c) 39 (d) 40
  23. 23.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 157 (c) 5 (d) 25
  24. 24.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  25. 25.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 181 (d) 1
  26. 26.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 7 (c) 8 (d) 6
  27. 27.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 73 (d) 71
  28. 28.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 148 (c) 196 (d) 172
  29. 29.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4366 (c) 4356 (d) 5476
  30. 30.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x − 4)(x − 7) (d) (x + 1)(x + 28)
  31. 31.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 2 (c) 4 (d) 256
  32. 32.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 16 (d) 63
  33. 33.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
  34. 34.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 12 (d) 10
  35. 35.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
  36. 36.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 1024 (d) 896
  37. 37.What are the roots of x² − 13x + 36 = 0? (a) 4 and 9 (b) 1 and 36 (c) 2 and 18 (d) 1 and 12
  38. 38.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 170 (c) 16 (d) 4
  39. 39.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 16 (c) 15 (d) 14
  40. 40.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 7 (c) 9 (d) 5

Answer key

  1. 121
  2. 20
  3. 289
  4. 25
  5. 49
  6. 4 and 5
  7. (x + 2)(x + 3)
  8. 181
  9. 74
  10. 9801
  11. 9
  12. 529
  13. 10609
  14. 15
  15. (x + 7)(x + 9)
  16. 3249
  17. (x + 3)(x + 4)
  18. 8281
  19. 441
  20. (x + 2)(x + 9)
  21. 15
  22. 40
  23. 25
  24. (x + 8)(x + 9)
  25. 1
  26. 7
  27. 72
  28. 148
  29. 4356
  30. (x + 4)(x + 7)
  31. 4
  32. 63
  33. (x + 6)(x + 7)
  34. 11
  35. (x + 2)(x + 4)
  36. 784
  37. 4 and 9
  38. 16
  39. 15
  40. 5

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