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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 959 · 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.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 11 (c) 29 (d) 30
  2. 2.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 133 (c) 144 (d) 166
  3. 3.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
  4. 4.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1521 (c) 1681 (d) 1531
  5. 5.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 2 and 10 (c) 1 and 20 (d) 1 and 8
  6. 6.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 14 (c) 40 (d) 13
  7. 7.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 65 (d) 177
  8. 8.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 9 (c) 20 (d) 10
  9. 9.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 100 (c) 52 (d) 148
  10. 10.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 25 (c) 73 (d) 49
  11. 11.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)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6084 (c) 6396 (d) 6724
  13. 13.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8281 (c) 8291 (d) 8101
  14. 14.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 537 (c) 361 (d) 273
  15. 15.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 7 (c) 13 (d) 12
  16. 16.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)
  17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
  18. 18.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 16 (d) 17
  19. 19.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 52 (c) 88 (d) 64
  20. 20.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  21. 21.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 91 (c) 1 (d) 61
  22. 22.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
  23. 23.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 63 (c) 45 (d) 117
  24. 24.If a + b = 7 and ab = 12, what is (a − b)²? (a) 37 (b) 25 (c) 1 (d) 49
  25. 25.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 337 (d) 225
  26. 26.If a + b = 16 and ab = 55, what is (a − b)²? (a) 36 (b) 6 (c) 146 (d) 256
  27. 27.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2219 (c) 2491 (d) 2209
  28. 28.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1156 (d) 1166
  29. 29.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 196 (c) 130 (d) 163
  30. 30.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1591 (c) 1379 (d) 1849
  31. 31.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 14 (d) 16
  32. 32.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)
  33. 33.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)
  34. 34.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
  35. 35.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 25 (c) 24 (d) 11
  36. 36.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3844 (c) 3596 (d) 3364
  37. 37.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 657 (d) 225
  38. 38.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 4 (d) 400
  39. 39.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 2 and 5 (c) 1 and 8 (d) 2 and 4
  40. 40.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56

Answer key

  1. 30
  2. 122
  3. (x + 4)(x + 6)
  4. 1521
  5. 4 and 5
  6. 13
  7. 65
  8. 9
  9. 52
  10. 25
  11. (x + 6)(x + 7)
  12. 6084
  13. 8281
  14. 185
  15. 12
  16. (x + 3)(x + 7)
  17. (x + 6)(x + 8)
  18. 17
  19. 40
  20. (x + 5)(x + 8)
  21. 1
  22. (x + 4)(x + 7)
  23. 45
  24. 1
  25. 113
  26. 36
  27. 2209
  28. 1156
  29. 130
  30. 1369
  31. 15
  32. (x + 8)(x + 9)
  33. (x + 2)(x + 6)
  34. (x + 3)(x + 6)
  35. 24
  36. 3364
  37. 225
  38. 4
  39. 2 and 4
  40. 56

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