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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 55 · 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.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 729 (c) 909 (d) 1089
  2. 2.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 15 (c) 14 (d) 16
  3. 3.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9604 (c) 9614 (d) 10404
  4. 4.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3609 (c) 3249 (d) 3979
  5. 5.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 130 (c) 163 (d) 196
  6. 6.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 177 (c) 93 (d) 121
  7. 7.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 45 (c) 81 (d) 3
  8. 8.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5476 (d) 5486
  9. 9.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 14 (c) 45 (d) 13
  10. 10.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 144 (c) 36 (d) 6
  11. 11.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 400 (c) 4 (d) 2
  12. 12.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 225 (d) 9
  13. 13.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  15. 15.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 4909 (c) 5329 (d) 5339
  16. 16.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 100 (c) 52 (d) 148
  17. 17.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 1 and 10 (d) 5 and 6
  18. 18.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 1 (c) 225 (d) 113
  19. 19.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
  20. 20.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8091 (c) 8649 (d) 7569
  21. 21.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 144 (c) 16 (d) 4
  22. 22.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 208 (c) 80 (d) 144
  23. 23.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1225 (c) 2025 (d) 1625
  24. 24.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 425 (c) 225 (d) 625
  25. 25.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 1 and 63 (d) 7 and 9
  26. 26.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 117 (c) 9 (d) 3
  27. 27.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 18 (d) 72
  28. 28.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2819 (c) 2509 (d) 2209
  29. 29.If a + b = 12 and ab = 20, what is (a − b)²? (a) 64 (b) 8 (c) 144 (d) 104
  30. 30.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
  31. 31.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 7744 (c) 8474 (d) 8104
  32. 32.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 8 (c) 4 and 5 (d) 1 and 20
  33. 33.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 24) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
  34. 34.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 21 (c) 9 (d) 20
  35. 35.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9801 (c) 10201 (d) 9999
  36. 36.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
  37. 37.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 34 (c) 94 (d) 64
  38. 38.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 189 (c) 297 (d) 153
  39. 39.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3136 (c) 4096 (d) 3584
  40. 40.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 148 (c) 244 (d) 172

Answer key

  1. 1089
  2. 15
  3. 9604
  4. 3969
  5. 130
  6. 65
  7. 9
  8. 5476
  9. 14
  10. 36
  11. 4
  12. 9
  13. (x + 6)(x + 7)
  14. (x + 7)(x + 8)
  15. 5329
  16. 52
  17. 5 and 6
  18. 1
  19. (x + 6)(x + 9)
  20. 7569
  21. 16
  22. 80
  23. 2025
  24. 625
  25. 7 and 9
  26. 9
  27. 17
  28. 2809
  29. 64
  30. (x + 2)(x + 4)
  31. 8464
  32. 4 and 5
  33. (x + 4)(x + 6)
  34. 20
  35. 9801
  36. (x + 3)(x + 8)
  37. 34
  38. 153
  39. 3136
  40. 148

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