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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 685 · 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.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)
  2. 2.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 81 (c) 73 (d) 97
  3. 3.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 9 (c) 17 (d) 25
  4. 4.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 63 (c) 17 (d) 16
  5. 5.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 40 (c) 52 (d) 88
  6. 6.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 55 (c) 57 (d) 15
  7. 7.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 262 (c) 130 (d) 196
  8. 8.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
  9. 9.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 2)(x + 6)
  10. 10.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
  11. 11.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)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4771 (d) 4761
  13. 13.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 325 (c) 125 (d) 175
  14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
  15. 15.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 6416 (c) 5776 (d) 7056
  16. 16.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 5)
  17. 17.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
  18. 18.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
  19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
  20. 20.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)
  21. 21.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 5 (c) 2 and 4 (d) 1 and 8
  22. 22.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 289 (c) 157 (d) 25
  23. 23.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 170 (c) 4 (d) 16
  24. 24.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
  25. 25.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
  26. 26.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4896 (c) 5184 (d) 4624
  27. 27.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 73 (c) 169 (d) 97
  28. 28.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
  29. 29.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 334 (c) 484 (d) 324
  30. 30.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 14 (c) 13 (d) 36
  31. 31.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 1 and 14 (c) 3 and 6 (d) 2 and 7
  32. 32.If a + b = 11 and ab = 18, what is (a − b)²? (a) 49 (b) 7 (c) 121 (d) 85
  33. 33.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3249 (c) 3979 (d) 3609
  34. 34.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 17 (c) 19 (d) 11
  35. 35.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 304 (c) 592 (d) 208
  36. 36.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 41 (d) 39
  37. 37.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 10) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
  38. 38.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 10 (c) 9 (d) 8
  39. 39.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 12 (d) 14
  40. 40.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 164 (c) 484 (d) 324

Answer key

  1. (x + 4)(x + 6)
  2. 65
  3. 9
  4. 16
  5. 40
  6. 56
  7. 130
  8. (x + 2)(x + 8)
  9. (x + 2)(x + 6)
  10. (x + 3)(x + 9)
  11. (x + 8)(x + 9)
  12. 4761
  13. 125
  14. (x + 7)(x + 8)
  15. 7056
  16. (x + 2)(x + 4)
  17. 17
  18. (x + 2)(x + 3)
  19. (x + 7)(x + 9)
  20. (x + 6)(x + 8)
  21. 2 and 4
  22. 25
  23. 16
  24. (x + 3)(x + 4)
  25. (x + 5)(x + 9)
  26. 4624
  27. 73
  28. (x + 6)(x + 7)
  29. 324
  30. 13
  31. 2 and 7
  32. 49
  33. 3969
  34. 18
  35. 208
  36. 40
  37. (x + 4)(x + 7)
  38. 9
  39. 13
  40. 164

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