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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 783 · 40 questions
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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 sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
  2. 2.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 36 (c) 6 (d) 64
  3. 3.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
  4. 4.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  5. 5.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
  6. 6.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 14 (d) 1 and 56
  7. 7.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 196 (c) 156 (d) 116
  8. 8.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  9. 9.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 10816 (c) 9216 (d) 9984
  10. 10.If a + b = 14 and ab = 24, what is a² + b²? (a) 148 (b) 172 (c) 196 (d) 244
  11. 11.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
  12. 12.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 76 (c) 100 (d) 52
  13. 13.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x − 6)(x − 9)
  14. 14.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 4 (c) 256 (d) 130
  15. 15.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 271 (c) 181 (d) 1
  16. 16.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 256 (c) 136 (d) 4
  17. 17.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
  18. 18.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  19. 19.If a + b = 7 and ab = 10, what is (a − b)²? (a) 3 (b) 29 (c) 49 (d) 9
  20. 20.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 4901 (c) 4761 (d) 5051
  21. 21.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 9 (c) 10 (d) 14
  22. 22.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  23. 23.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 15 (c) 14 (d) 13
  24. 24.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 9 (c) 5 (d) 7
  25. 25.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 401 (c) 361 (d) 441
  26. 26.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
  27. 27.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 14 (c) 15 (d) 13
  28. 28.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 1)(x + 35)
  29. 29.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 225 (d) 169
  30. 30.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
  31. 31.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 16 (c) 400 (d) 4
  32. 32.What are the roots of x² − 7x + 10 = 0? (a) 2 and 5 (b) 1 and 10 (c) 1 and 6 (d) 3 and 4
  33. 33.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 65 (c) 177 (d) 121
  34. 34.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 5776 (c) 7066 (d) 6416
  35. 35.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)
  36. 36.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 537 (d) 361
  37. 37.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 17 (c) 9 (d) 25
  38. 38.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 2 and 10 (c) 1 and 35 (d) 1 and 11
  39. 39.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 225 (c) 1 (d) 113
  40. 40.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)

Answer key

  1. 17
  2. 36
  3. (x + 5)(x + 6)
  4. (x + 7)(x + 9)
  5. (x + 2)(x + 4)
  6. 7 and 8
  7. 116
  8. (x + 6)(x + 7)
  9. 9216
  10. 148
  11. (x + 6)(x + 8)
  12. 52
  13. (x + 6)(x + 9)
  14. 4
  15. 1
  16. 16
  17. (x + 2)(x + 6)
  18. (x + 8)(x + 9)
  19. 9
  20. 5041
  21. 9
  22. (x + 4)(x + 8)
  23. 14
  24. 1
  25. 441
  26. (x + 3)(x + 7)
  27. 14
  28. (x + 5)(x + 7)
  29. 113
  30. (x + 5)(x + 9)
  31. 16
  32. 2 and 5
  33. 65
  34. 7056
  35. (x + 7)(x + 8)
  36. 185
  37. 9
  38. 5 and 7
  39. 1
  40. (x + 3)(x + 4)

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