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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 406 · 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 = 8 and ab = 12, what is a² + b²? (a) 64 (b) 52 (c) 40 (d) 88
  2. 2.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
  3. 3.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4761 (c) 4901 (d) 5041
  4. 4.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 5329 (c) 4489 (d) 4891
  5. 5.If a + b = 8 and ab = 7, what is (a − b)²? (a) 6 (b) 36 (c) 50 (d) 64
  6. 6.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 10404 (d) 9604
  7. 7.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 916 (c) 1156 (d) 676
  8. 8.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  9. 9.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 5 (c) 73 (d) 121
  10. 10.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 17 (d) 63
  11. 11.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 139 (c) 229 (d) 109
  12. 12.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 16 (c) 14 (d) 8
  13. 13.If a + b = 10 and ab = 9, what is (a − b)²? (a) 8 (b) 64 (c) 100 (d) 82
  14. 14.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x − 2)(x − 8)
  15. 15.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 2 (c) 100 (d) 196
  16. 16.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 361 (d) 271
  17. 17.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 333 (c) 225 (d) 117
  18. 18.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3721 (c) 3731 (d) 3601
  19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  20. 20.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x + 1)(x + 12) (d) (x − 3)(x − 4)
  21. 21.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
  22. 22.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 9 (c) 8 (d) 20
  23. 23.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  24. 24.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 1 (c) 225 (d) 113
  25. 25.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
  26. 26.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 10 (d) 9
  27. 27.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 19 (c) 9 (d) 21
  28. 28.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 112 (c) 144 (d) 80
  29. 29.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 5 and 6 (d) 1 and 30
  30. 30.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 144 (c) 4 (d) 2
  31. 31.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 130 (c) 196 (d) 262
  32. 32.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 484 (c) 396 (d) 324
  33. 33.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 7 (c) 13 (d) 9
  34. 34.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1531 (c) 1521 (d) 1599
  35. 35.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3596 (c) 3844 (d) 3364
  36. 36.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 185 (d) 273
  37. 37.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 8 (d) 7
  38. 38.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 17 (d) 18
  39. 39.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 17 (d) 71
  40. 40.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1691 (c) 1601 (d) 1521

Answer key

  1. 40
  2. (x + 3)(x + 9)
  3. 5041
  4. 4489
  5. 36
  6. 10404
  7. 1156
  8. (x + 5)(x + 8)
  9. 25
  10. 16
  11. 109
  12. 15
  13. 64
  14. (x + 2)(x + 8)
  15. 4
  16. 1
  17. 117
  18. 3721
  19. (x + 8)(x + 9)
  20. (x + 3)(x + 4)
  21. (x + 6)(x + 7)
  22. 9
  23. (x + 7)(x + 8)
  24. 1
  25. (x + 3)(x + 6)
  26. 10
  27. 20
  28. 80
  29. 5 and 6
  30. 4
  31. 130
  32. 324
  33. 5
  34. 1521
  35. 3364
  36. 185
  37. 6
  38. 17
  39. 72
  40. 1681

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