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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 517 · 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 = 12 and ab = 27, what is a² + b²? (a) 198 (b) 117 (c) 144 (d) 90
  2. 2.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 9)
  3. 3.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  4. 4.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 100 (c) 144 (d) 122
  5. 5.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
  6. 6.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3364 (c) 3604 (d) 3854
  7. 7.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 121 (c) 1 (d) 61
  8. 8.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 5329 (c) 4489 (d) 4891
  9. 9.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)
  10. 10.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 80 (c) 16 (d) 4
  11. 11.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 27 (c) 12 (d) 26
  12. 12.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  13. 13.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)
  14. 14.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 169 (c) 253 (d) 85
  15. 15.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 11035 (c) 9025 (d) 10025
  16. 16.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 37 (c) 49 (d) 1
  17. 17.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 484 (d) 244
  18. 18.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 229 (c) 289 (d) 169
  19. 19.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2219 (c) 2491 (d) 2809
  20. 20.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 896 (c) 794 (d) 1024
  21. 21.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 35 (c) 1 and 11 (d) 5 and 7
  22. 22.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5776 (b) 6384 (c) 5786 (d) 7056
  23. 23.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2126 (c) 2916 (d) 2116
  24. 24.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 625 (c) 1225 (d) 925
  25. 25.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 361 (d) 371
  26. 26.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
  27. 27.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 4)(x + 9) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
  28. 28.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 361 (d) 271
  29. 29.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 145 (c) 217 (d) 289
  30. 30.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 52 (c) 100 (d) 2
  31. 31.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 324 (c) 2 (d) 4
  32. 32.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
  33. 33.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9409 (c) 9991 (d) 9419
  34. 34.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1296 (c) 1936 (d) 1306
  35. 35.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 5184 (c) 4634 (d) 4624
  36. 36.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 169 (c) 1 (d) 127
  37. 37.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x − 4)(x − 8)
  38. 38.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 73 (c) 25 (d) 49
  39. 39.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 441 (c) 225 (d) 9
  40. 40.What is the product of the roots of x² − 12x + 32 = 0? (a) 32 (b) 33 (c) 31 (d) 12

Answer key

  1. 90
  2. (x + 3)(x + 7)
  3. (x + 6)(x + 8)
  4. 100
  5. (x + 3)(x + 9)
  6. 3844
  7. 1
  8. 4489
  9. (x + 2)(x + 4)
  10. 16
  11. 27
  12. (x + 5)(x + 7)
  13. (x + 8)(x + 9)
  14. 85
  15. 11025
  16. 1
  17. 164
  18. 169
  19. 2209
  20. 784
  21. 5 and 7
  22. 5776
  23. 2116
  24. 1225
  25. 361
  26. (x + 4)(x + 7)
  27. (x + 4)(x + 9)
  28. 181
  29. 145
  30. 4
  31. 4
  32. (x + 5)(x + 6)
  33. 9409
  34. 1296
  35. 4624
  36. 1
  37. (x + 4)(x + 8)
  38. 25
  39. 9
  40. 32

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