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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 30 · 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 + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2411 (c) 2601 (d) 2499
  3. 3.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8109 (c) 8659 (d) 7569
  4. 4.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 9801 (c) 10201 (d) 10001
  5. 5.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9999 (c) 9811 (d) 10201
  6. 6.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 3 and 7 (c) 1 and 16 (d) 2 and 8
  7. 7.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 1089 (c) 909 (d) 729
  8. 8.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
  9. 9.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 17 (d) 19
  10. 10.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 144 (c) 4 (d) 16
  11. 11.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  12. 12.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 11 (c) 10 (d) 18
  13. 13.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 7744 (c) 8464 (d) 7754
  14. 14.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 65 (c) 73 (d) 97
  15. 15.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 193 (c) 361 (d) 25
  16. 16.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 18 (c) 10 (d) 9
  17. 17.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 3 (c) 9 (d) 225
  18. 18.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (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.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5041 (c) 4761 (d) 5051
  21. 21.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 484 (d) 244
  22. 22.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 34 (c) 2 (d) 4
  23. 23.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)
  24. 24.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 97 (d) 121
  25. 25.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
  26. 26.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 16 (c) 10 (d) 22
  27. 27.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 13 (d) 12
  28. 28.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
  29. 29.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 21 (c) 10 (d) 20
  30. 30.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 63 (d) 17
  31. 31.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
  32. 32.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2714 (c) 2304 (d) 2504
  33. 33.If a + b = 19 and ab = 84, what is a² + b²? (a) 193 (b) 277 (c) 361 (d) 529
  34. 34.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6084 (c) 6724 (d) 6404
  35. 35.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 125 (c) 225 (d) 5
  36. 36.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 8 (c) 7 (d) 6
  37. 37.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
  38. 38.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 1 (d) 169
  39. 39.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 53 (d) 55
  40. 40.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16

Answer key

  1. (x + 3)(x + 6)
  2. 2401
  3. 8649
  4. 10201
  5. 9801
  6. 2 and 8
  7. 1089
  8. (x + 5)(x + 9)
  9. 18
  10. 16
  11. (x + 6)(x + 8)
  12. 11
  13. 7744
  14. 65
  15. 25
  16. 9
  17. 9
  18. (x + 2)(x + 3)
  19. (x + 7)(x + 9)
  20. 5041
  21. 164
  22. 4
  23. (x + 5)(x + 6)
  24. 73
  25. (x + 5)(x + 8)
  26. 10
  27. 13
  28. (x + 3)(x + 5)
  29. 21
  30. 16
  31. (x + 8)(x + 9)
  32. 2704
  33. 193
  34. 6724
  35. 25
  36. 6
  37. 8 and 9
  38. 1
  39. 54
  40. 17

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