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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 578 · 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 = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 148 (c) 10 (d) 100
  2. 2.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 74 (c) 214 (d) 144
  3. 3.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 361 (d) 271
  4. 4.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 324 (c) 4 (d) 164
  5. 5.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 784 (c) 1034 (d) 904
  6. 6.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 1 and 7 (c) 2 and 6 (d) 1 and 15
  7. 7.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 4)(x + 7) (c) (x − 4)(x − 7) (d) (x + 1)(x + 10)
  8. 8.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  9. 9.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 1 (c) 127 (d) 85
  10. 10.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 65 (c) 73 (d) 97
  11. 11.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 14 (d) 48
  12. 12.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 256 (c) 130 (d) 2
  13. 13.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  14. 14.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 6) (d) (x + 1)(x + 10)
  15. 15.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 4 (d) 16
  16. 16.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 100 (c) 52 (d) 2
  17. 17.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 80 (c) 4 (d) 16
  18. 18.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 12 (c) 18 (d) 10
  19. 19.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10001 (c) 10211 (d) 10201
  20. 20.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
  21. 21.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 28 (c) 36 (d) 20
  22. 22.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 382 (c) 130 (d) 193
  23. 23.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 144 (c) 6 (d) 90
  24. 24.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 4761 (c) 4901 (d) 5051
  25. 25.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 277 (c) 361 (d) 529
  26. 26.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 625 (c) 225 (d) 235
  27. 27.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 181 (c) 91 (d) 61
  28. 28.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 7744 (c) 8474 (d) 8104
  29. 29.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2304 (c) 2714 (d) 2704
  30. 30.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 121 (c) 111 (d) 101
  31. 31.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 12 (c) 1 and 40 (d) 2 and 20
  32. 32.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
  33. 33.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  34. 34.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 9 (d) 3
  35. 35.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1681 (c) 1531 (d) 1599
  36. 36.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)
  37. 37.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 3 (c) 361 (d) 185
  38. 38.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 12 (d) 11
  39. 39.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 169 (c) 9 (d) 81
  40. 40.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)

Answer key

  1. 100
  2. 74
  3. 1
  4. 4
  5. 1024
  6. 3 and 5
  7. (x + 4)(x + 7)
  8. (x + 6)(x + 8)
  9. 1
  10. 65
  11. 48
  12. 4
  13. (x + 8)(x + 9)
  14. (x + 2)(x + 5)
  15. 16
  16. 4
  17. 16
  18. 11
  19. 10201
  20. (x + 3)(x + 4)
  21. 20
  22. 130
  23. 36
  24. 5041
  25. 193
  26. 225
  27. 61
  28. 8464
  29. 2704
  30. 101
  31. 5 and 8
  32. (x + 3)(x + 8)
  33. 17
  34. 9
  35. 1521
  36. (x + 2)(x + 8)
  37. 9
  38. 11
  39. 81
  40. (x + 7)(x + 9)

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