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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 738 · 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 = 19 and ab = 88, what is a² + b²? (a) 537 (b) 273 (c) 361 (d) 185
  2. 2.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  3. 3.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 133 (c) 241 (d) 97
  4. 4.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 25 (c) 24 (d) 23
  5. 5.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 3 (c) 9 (d) 81
  6. 6.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 39 (c) 41 (d) 13
  7. 7.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 7 (c) 5 (d) 6
  8. 8.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6241 (d) 6251
  9. 9.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 208 (c) 80 (d) 112
  10. 10.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 89 (c) 169 (d) 3
  11. 11.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 62 (c) 64 (d) 63
  12. 12.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 4 (c) 20 (d) 36
  13. 13.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  14. 14.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 10816 (c) 9216 (d) 10826
  15. 15.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 361 (c) 193 (d) 5
  16. 16.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  17. 17.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 2025 (c) 1575 (d) 1225
  18. 18.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 16) (d) (x + 1)(x + 9)
  19. 19.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8281 (c) 8099 (d) 7931
  20. 20.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 202 (c) 400 (d) 301
  21. 21.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9811 (c) 10201 (d) 9999
  22. 22.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
  23. 23.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 25 (c) 5 (d) 169
  24. 24.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)
  25. 25.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 144 (c) 109 (d) 74
  26. 26.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8836 (c) 8116 (d) 8846
  27. 27.If a + b = 16 and ab = 60, what is (a − b)²? (a) 256 (b) 16 (c) 4 (d) 136
  28. 28.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 121 (c) 181 (d) 91
  29. 29.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 9 (c) 153 (d) 225
  30. 30.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 56 (c) 57 (d) 15
  31. 31.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
  32. 32.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10414 (c) 10404 (d) 9604
  34. 34.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 12)
  35. 35.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
  36. 36.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 10025 (c) 11035 (d) 11025
  37. 37.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 15 (d) 13
  38. 38.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 1 and 30 (d) 5 and 6
  39. 39.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
  40. 40.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 10 (c) 6 (d) 7

Answer key

  1. 185
  2. (x + 6)(x + 7)
  3. 97
  4. 24
  5. 9
  6. 40
  7. 6
  8. 6241
  9. 80
  10. 9
  11. 63
  12. 4
  13. (x + 7)(x + 8)
  14. 10816
  15. 25
  16. (x + 8)(x + 9)
  17. 1225
  18. (x + 2)(x + 8)
  19. 7921
  20. 202
  21. 9801
  22. 17
  23. 25
  24. (x + 5)(x + 6)
  25. 74
  26. 8836
  27. 16
  28. 61
  29. 81
  30. 56
  31. (x + 4)(x + 9)
  32. (x + 3)(x + 8)
  33. 10404
  34. (x + 2)(x + 6)
  35. (x + 4)(x + 8)
  36. 11025
  37. 14
  38. 5 and 6
  39. (x + 4)(x + 6)
  40. 7

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