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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 224 · 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 = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 289 (c) 145 (d) 1
  2. 2.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2499 (c) 2401 (d) 2411
  3. 3.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 5776 (c) 6384 (d) 7056
  4. 4.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 5041 (c) 4899 (d) 4761
  5. 5.If a + b = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 16 (c) 64 (d) 4
  6. 6.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
  7. 7.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)
  8. 8.If a + b = 14 and ab = 40, what is (a − b)²? (a) 36 (b) 116 (c) 196 (d) 6
  9. 9.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 53 (d) 55
  10. 10.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 37 (c) 13 (d) 19
  11. 11.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 55 (d) 15
  12. 12.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1681 (c) 1599 (d) 1531
  13. 13.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 103 (c) 157 (d) 85
  14. 14.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 971 (c) 961 (d) 901
  15. 15.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
  16. 16.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 441 (d) 333
  17. 17.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 468 (c) 324 (d) 252
  18. 18.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 851 (c) 841 (d) 899
  19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
  20. 20.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 50 (c) 57 (d) 64
  21. 21.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 2 and 4 (c) 1 and 8 (d) 2 and 5
  22. 22.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 121 (c) 73 (d) 5
  23. 23.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
  24. 24.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 10 (c) 9 (d) 24
  25. 25.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 3 (c) 289 (d) 149
  26. 26.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 14 (d) 15
  27. 27.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 4 (c) 20 (d) 36
  28. 28.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 494 (d) 484
  29. 29.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 14 (d) 15
  30. 30.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5329 (b) 5339 (c) 4489 (d) 4909
  31. 31.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9604 (c) 10404 (d) 9614
  32. 32.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 64 (c) 94 (d) 34
  33. 33.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 12 (c) 34 (d) 35
  34. 34.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 366 (c) 256 (d) 146
  35. 35.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1521 (c) 1601 (d) 1691
  36. 36.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 42 (c) 1 and 12 (d) 6 and 7
  37. 37.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)
  38. 38.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  39. 39.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3721 (b) 3491 (c) 3481 (d) 3599
  40. 40.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 58 (c) 100 (d) 79

Answer key

  1. 1
  2. 2401
  3. 5776
  4. 4761
  5. 16
  6. (x + 4)(x + 6)
  7. (x + 7)(x + 8)
  8. 36
  9. 54
  10. 13
  11. 56
  12. 1521
  13. 85
  14. 961
  15. (x + 4)(x + 9)
  16. 225
  17. 180
  18. 841
  19. (x + 7)(x + 9)
  20. 50
  21. 2 and 4
  22. 25
  23. 181
  24. 10
  25. 9
  26. 15
  27. 4
  28. 484
  29. 14
  30. 5329
  31. 9604
  32. 34
  33. 35
  34. 146
  35. 1681
  36. 6 and 7
  37. (x + 6)(x + 8)
  38. (x + 6)(x + 7)
  39. 3481
  40. 58

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