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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 478 · 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 52²? (a) 2714 (b) 2304 (c) 2704 (d) 2504
  2. 2.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 15 (c) 54 (d) 53
  3. 3.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  4. 4.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  5. 5.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 3 and 5 (c) 1 and 15 (d) 2 and 6
  6. 6.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 20) (d) (x + 1)(x + 8)
  7. 7.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5184 (b) 4904 (c) 4624 (d) 5194
  8. 8.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 116 (c) 276 (d) 156
  9. 9.If a + b = 10 and ab = 24, what is a² + b²? (a) 52 (b) 100 (c) 76 (d) 148
  10. 10.If a + b = 13 and ab = 12, what is a² + b²? (a) 157 (b) 169 (c) 145 (d) 193
  11. 11.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 8075 (c) 9025 (d) 7235
  12. 12.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
  13. 13.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3491 (c) 3599 (d) 3481
  14. 14.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 85 (c) 253 (d) 127
  15. 15.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 185 (c) 361 (d) 3
  16. 16.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 19 (d) 20
  17. 17.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 164 (c) 324 (d) 484
  18. 18.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 13 (c) 42 (d) 12
  19. 19.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7235 (c) 7225 (d) 5625
  20. 20.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 625 (c) 925 (d) 1235
  21. 21.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
  22. 22.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x + 1)(x + 10)
  23. 23.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
  24. 24.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 2 and 8 (c) 3 and 7 (d) 1 and 9
  25. 25.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 42 (d) 13
  26. 26.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  27. 27.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)
  28. 28.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 73 (d) 71
  29. 29.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
  30. 30.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1531 (c) 1599 (d) 1521
  31. 31.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 29 (c) 3 (d) 9
  32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
  33. 33.If a + b = 14 and ab = 48, what is a² + b²? (a) 148 (b) 100 (c) 196 (d) 292
  34. 34.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 256 (c) 130 (d) 193
  35. 35.If a + b = 18 and ab = 72, what is a² + b²? (a) 252 (b) 468 (c) 180 (d) 324
  36. 36.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 13 (c) 19 (d) 1
  37. 37.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 537 (d) 361
  38. 38.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 68 (c) 132 (d) 100
  39. 39.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 97 (c) 5 (d) 25
  40. 40.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)

Answer key

  1. 2704
  2. 54
  3. (x + 6)(x + 7)
  4. (x + 5)(x + 8)
  5. 3 and 5
  6. (x + 4)(x + 5)
  7. 5184
  8. 116
  9. 52
  10. 145
  11. 7225
  12. (x + 2)(x + 8)
  13. 3481
  14. 85
  15. 9
  16. 20
  17. 164
  18. 13
  19. 7225
  20. 1225
  21. (x + 3)(x + 6)
  22. (x + 4)(x + 7)
  23. (x + 3)(x + 8)
  24. 2 and 8
  25. 42
  26. (x + 8)(x + 9)
  27. (x + 7)(x + 8)
  28. 72
  29. 8 and 9
  30. 1521
  31. 9
  32. 17
  33. 100
  34. 130
  35. 180
  36. 1
  37. 185
  38. 68
  39. 25
  40. (x + 6)(x + 8)

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