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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 733 · 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 + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 20)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 7225 (c) 6425 (d) 5625
  3. 3.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 18 (c) 8 (d) 9
  4. 4.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
  5. 5.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2209 (c) 2809 (d) 2219
  6. 6.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 409 (c) 169 (d) 229
  7. 7.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 81 (c) 1 (d) 41
  8. 8.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 6375 (b) 5625 (c) 5635 (d) 7225
  9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  10. 10.If a + b = 14 and ab = 33, what is (a − b)²? (a) 8 (b) 130 (c) 196 (d) 64
  11. 11.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 25 (c) 225 (d) 125
  12. 12.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 64 (d) 16
  13. 13.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 400 (c) 598 (d) 301
  14. 14.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 181 (c) 137 (d) 313
  15. 15.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
  16. 16.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 729 (c) 909 (d) 1099
  18. 18.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 891 (c) 1089 (d) 729
  19. 19.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)
  20. 20.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
  21. 21.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 494 (c) 324 (d) 484
  22. 22.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 81 (c) 3 (d) 9
  23. 23.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 130 (d) 256
  24. 24.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 164 (d) 324
  25. 25.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 170 (c) 4 (d) 324
  26. 26.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1849 (c) 1609 (d) 1859
  27. 27.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 64 (c) 49 (d) 34
  28. 28.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
  29. 29.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 74 (c) 109 (d) 144
  30. 30.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 256 (c) 208 (d) 352
  31. 31.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 2 and 5 (c) 1 and 6 (d) 1 and 10
  32. 32.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 361 (c) 271 (d) 181
  33. 33.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 6 (c) 7 (d) 12
  34. 34.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 8099 (c) 7921 (d) 7931
  35. 35.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 841 (c) 971 (d) 901
  36. 36.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
  37. 37.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
  38. 38.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 36 (c) 37 (d) 35
  39. 39.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
  40. 40.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 1 (c) 5 (d) 7

Answer key

  1. (x + 4)(x + 5)
  2. 7225
  3. 9
  4. (x + 6)(x + 8)
  5. 2209
  6. 169
  7. 1
  8. 5625
  9. (x + 8)(x + 9)
  10. 64
  11. 25
  12. 63
  13. 202
  14. 137
  15. (x + 5)(x + 9)
  16. (x + 4)(x + 6)
  17. 1089
  18. 729
  19. (x + 3)(x + 6)
  20. (x + 5)(x + 7)
  21. 484
  22. 9
  23. 130
  24. 164
  25. 16
  26. 1849
  27. 34
  28. (x + 2)(x + 7)
  29. 74
  30. 160
  31. 2 and 5
  32. 1
  33. 7
  34. 7921
  35. 961
  36. 17
  37. (x + 3)(x + 7)
  38. 36
  39. 145
  40. 1

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