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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 475 · 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 98²? (a) 9604 (b) 10404 (c) 9614 (d) 9996
  2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  3. 3.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 181 (c) 1 (d) 361
  4. 4.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  5. 5.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 324 (c) 6 (d) 36
  6. 6.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6571 (d) 6561
  7. 7.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 1 and 12 (c) 2 and 6 (d) 3 and 4
  8. 8.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
  9. 9.What is the product of the roots of x² − 10x + 16 = 0? (a) 10 (b) 15 (c) 17 (d) 16
  10. 10.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2304 (d) 2314
  11. 11.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 16 (c) 4 (d) 80
  12. 12.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 841 (c) 961 (d) 851
  13. 13.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 63 (d) 16
  14. 14.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 12 (c) 13 (d) 11
  15. 15.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
  16. 16.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1616 (d) 1936
  17. 17.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 57 (d) 55
  18. 18.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 7 (c) 5 (d) 1
  19. 19.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 31 (c) 33 (d) 32
  20. 20.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 10016 (c) 10826 (d) 9216
  21. 21.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8836 (c) 7396 (d) 8084
  22. 22.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 18 (c) 1 and 8 (d) 3 and 6
  23. 23.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 72 (d) 73
  24. 24.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
  25. 25.If a + b = 13 and ab = 40, what is (a − b)²? (a) 89 (b) 169 (c) 9 (d) 3
  26. 26.What is the product of the roots of x² − 7x + 12 = 0? (a) 13 (b) 7 (c) 12 (d) 11
  27. 27.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 15 (c) 14 (d) 45
  28. 28.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 2 and 8 (c) 1 and 21 (d) 3 and 7
  29. 29.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 13 (d) 48
  30. 30.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 625 (c) 225 (d) 425
  31. 31.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 25 (c) 361 (d) 5
  32. 32.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
  33. 33.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
  34. 34.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 3 and 15 (c) 1 and 13 (d) 1 and 45
  35. 35.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3249 (c) 3979 (d) 3609
  36. 36.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 225 (c) 9 (d) 81
  37. 37.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 10 (c) 12 (d) 11
  38. 38.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 5625 (d) 7225
  39. 39.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
  40. 40.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 151 (c) 196 (d) 106

Answer key

  1. 9604
  2. 17
  3. 1
  4. (x + 6)(x + 8)
  5. 36
  6. 6561
  7. 3 and 4
  8. (x + 3)(x + 4)
  9. 16
  10. 2304
  11. 16
  12. 841
  13. 63
  14. 12
  15. (x + 5)(x + 6)
  16. 1936
  17. 56
  18. 1
  19. 32
  20. 10816
  21. 7396
  22. 3 and 6
  23. 72
  24. (x + 3)(x + 8)
  25. 9
  26. 12
  27. 14
  28. 3 and 7
  29. 14
  30. 625
  31. 25
  32. (x + 8)(x + 9)
  33. (x + 7)(x + 8)
  34. 5 and 9
  35. 3969
  36. 81
  37. 11
  38. 7225
  39. (x + 6)(x + 9)
  40. 106

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