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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 296 · 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 is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
  2. 2.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1296 (c) 1616 (d) 1936
  4. 4.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 157 (c) 223 (d) 289
  5. 5.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 202 (d) 400
  6. 6.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 61 (c) 41 (d) 81
  7. 7.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 1089 (c) 909 (d) 729
  8. 8.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)
  9. 9.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 2)(x + 6) (c) (x + 1)(x + 7) (d) (x − 2)(x − 6)
  10. 10.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)
  11. 11.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
  12. 12.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
  13. 13.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 1 (c) 217 (d) 289
  14. 14.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
  15. 15.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
  16. 16.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 28 (d) 11
  17. 17.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 7 (c) 49 (d) 137
  18. 18.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 12 (c) 11 (d) 24
  19. 19.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 1 and 36 (c) 2 and 18 (d) 4 and 9
  20. 20.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)
  21. 21.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 304 (c) 400 (d) 208
  22. 22.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
  23. 23.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 14 (c) 13 (d) 36
  24. 24.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 324 (c) 252 (d) 180
  25. 25.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 52 (c) 2 (d) 100
  26. 26.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 4 (c) 1 and 6 (d) 2 and 4
  27. 27.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 352 (c) 160 (d) 208
  28. 28.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 9 (c) 225 (d) 153
  29. 29.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 225 (c) 153 (d) 189
  30. 30.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 784 (c) 904 (d) 1034
  31. 31.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 63 (d) 62
  32. 32.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 625 (c) 1235 (d) 925
  33. 33.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 271 (d) 541
  34. 34.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 494 (c) 324 (d) 404
  35. 35.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 4 (c) 144 (d) 2
  36. 36.If a + b = 10 and ab = 9, what is (a − b)²? (a) 64 (b) 82 (c) 8 (d) 100
  37. 37.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2209 (c) 2509 (d) 2809
  38. 38.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 409 (c) 229 (d) 169
  39. 39.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
  40. 40.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 3 and 8 (d) 1 and 10

Answer key

  1. 17
  2. (x + 3)(x + 4)
  3. 1936
  4. 157
  5. 4
  6. 1
  7. 1089
  8. (x + 7)(x + 9)
  9. (x + 2)(x + 6)
  10. (x + 8)(x + 9)
  11. (x + 4)(x + 5)
  12. (x + 6)(x + 8)
  13. 1
  14. (x + 3)(x + 8)
  15. (x + 3)(x + 9)
  16. 28
  17. 49
  18. 11
  19. 4 and 9
  20. (x + 2)(x + 3)
  21. 208
  22. (x + 5)(x + 9)
  23. 13
  24. 180
  25. 4
  26. 2 and 3
  27. 160
  28. 81
  29. 153
  30. 1024
  31. 63
  32. 1225
  33. 181
  34. 484
  35. 4
  36. 64
  37. 2809
  38. 169
  39. 2 and 7
  40. 3 and 8

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