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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 308 · 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 − 17x + 72 = 0? (a) 17 (b) 71 (c) 73 (d) 72
  2. 2.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 169 (c) 9 (d) 89
  3. 3.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 12 (c) 11 (d) 32
  4. 4.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 24 (c) 9 (d) 10
  5. 5.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15
  6. 6.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
  7. 7.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 62 (d) 63
  8. 8.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 7) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
  9. 9.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 2)(x + 9) (c) (x + 1)(x + 18) (d) (x + 1)(x + 10)
  10. 10.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 2 and 24 (c) 1 and 48 (d) 6 and 8
  11. 11.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 3 and 6 (c) 2 and 9 (d) 1 and 8
  12. 12.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 325 (c) 225 (d) 125
  13. 13.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 841 (c) 961 (d) 899
  14. 14.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4771 (c) 5041 (d) 4899
  15. 15.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11025 (c) 10025 (d) 11035
  16. 16.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 10816 (c) 9984 (d) 9216
  17. 17.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 36 (c) 12 (d) 13
  18. 18.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
  19. 19.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 425 (d) 625
  20. 20.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 11 (c) 12 (d) 30
  21. 21.If a + b = 15 and ab = 56, what is (a − b)²? (a) 113 (b) 225 (c) 169 (d) 1
  22. 22.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
  23. 23.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3596 (c) 3364 (d) 3374
  24. 24.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 16 (c) 400 (d) 208
  25. 25.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
  26. 26.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 85 (c) 121 (d) 49
  27. 27.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 576 (d) 586
  28. 28.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 5 (c) 7 (d) 6
  29. 29.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 333 (d) 441
  30. 30.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
  31. 31.If a + b = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 64 (c) 6 (d) 50
  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.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5184 (c) 5194 (d) 4904
  34. 34.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 217 (d) 145
  35. 35.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 34 (c) 12 (d) 35
  36. 36.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  37. 37.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 57 (d) 56
  38. 38.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17
  39. 39.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 361 (c) 537 (d) 273
  40. 40.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)

Answer key

  1. 72
  2. 9
  3. 12
  4. 10
  5. 15
  6. (x + 5)(x + 9)
  7. 63
  8. (x + 2)(x + 6)
  9. (x + 2)(x + 9)
  10. 6 and 8
  11. 3 and 6
  12. 125
  13. 841
  14. 4761
  15. 11025
  16. 9216
  17. 13
  18. (x + 4)(x + 7)
  19. 625
  20. 11
  21. 1
  22. (x + 3)(x + 6)
  23. 3364
  24. 16
  25. (x + 4)(x + 6)
  26. 49
  27. 576
  28. 5
  29. 225
  30. (x + 6)(x + 8)
  31. 36
  32. (x + 8)(x + 9)
  33. 5184
  34. 145
  35. 35
  36. (x + 7)(x + 8)
  37. 56
  38. 17
  39. 185
  40. (x + 4)(x + 5)

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