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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 102 · 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.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
  2. 2.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 337 (d) 225
  3. 3.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 4235 (c) 4225 (d) 5625
  4. 4.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9614 (c) 9996 (d) 9604
  5. 5.What is the sum of the roots of x² − 8x + 15 = 0? (a) 8 (b) 9 (c) 15 (d) 7
  6. 6.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181
  7. 7.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 10 (c) 100 (d) 196
  8. 8.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 277 (c) 529 (d) 361
  9. 9.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 229 (c) 109 (d) 169
  10. 10.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 181 (d) 137
  11. 11.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  12. 12.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
  13. 13.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
  14. 14.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 15 (d) 16
  15. 15.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 30 (c) 11 (d) 31
  16. 16.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 324 (c) 6 (d) 180
  17. 17.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1691 (c) 1681 (d) 1521
  18. 18.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 5 (c) 7 (d) 6
  19. 19.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 256 (c) 146 (d) 201
  20. 20.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 40 (c) 64 (d) 88
  21. 21.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 19 (c) 9 (d) 21
  22. 22.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 169 (c) 11 (d) 121
  23. 23.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 576 (c) 256 (d) 416
  24. 24.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 55 (d) 56
  25. 25.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 9 (d) 225
  26. 26.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)
  27. 27.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
  28. 28.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 169 (c) 193 (d) 145
  29. 29.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
  30. 30.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 9216 (c) 10826 (d) 10816
  31. 31.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 961 (c) 971 (d) 901
  32. 32.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 289 (d) 1
  33. 33.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 2025 (c) 3035 (d) 2525
  34. 34.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 89 (c) 169 (d) 9
  35. 35.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
  36. 36.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1235 (d) 1575
  37. 37.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
  38. 38.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 256 (c) 352 (d) 160
  39. 39.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 376 (c) 136 (d) 196
  40. 40.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 121 (c) 181 (d) 91

Answer key

  1. (x + 7)(x + 9)
  2. 113
  3. 4225
  4. 9604
  5. 8
  6. 181
  7. 100
  8. 193
  9. 109
  10. 137
  11. (x + 5)(x + 8)
  12. (x + 2)(x + 8)
  13. (x + 4)(x + 6)
  14. 16
  15. 30
  16. 36
  17. 1681
  18. 5
  19. 146
  20. 40
  21. 20
  22. 121
  23. 576
  24. 56
  25. 9
  26. (x + 6)(x + 8)
  27. (x + 6)(x + 9)
  28. 145
  29. (x + 2)(x + 3)
  30. 10816
  31. 961
  32. 1
  33. 3025
  34. 9
  35. (x + 5)(x + 9)
  36. 1225
  37. (x + 7)(x + 8)
  38. 160
  39. 136
  40. 61

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