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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 417 · 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 + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
  2. 2.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
  3. 3.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
  4. 4.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 396 (c) 334 (d) 484
  5. 5.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 170 (c) 478 (d) 324
  6. 6.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 34 (c) 49 (d) 64
  7. 7.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
  8. 8.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 7 (c) 12 (d) 6
  9. 9.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4891 (c) 4499 (d) 5329
  10. 10.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
  11. 11.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 256 (c) 384 (d) 576
  13. 13.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 1 and 72 (d) 8 and 9
  14. 14.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 185 (c) 9 (d) 361
  15. 15.If a + b = 11 and ab = 10, what is (a − b)²? (a) 121 (b) 9 (c) 101 (d) 81
  16. 16.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
  17. 17.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
  18. 18.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 4 (c) 64 (d) 2
  19. 19.What is the product of the roots of x² − 8x + 15 = 0? (a) 14 (b) 16 (c) 15 (d) 8
  20. 20.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 14 (d) 13
  21. 21.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 17 (c) 9 (d) 18
  22. 22.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 2025 (d) 1575
  23. 23.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 8 (c) 64 (d) 160
  24. 24.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
  25. 25.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
  26. 26.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 541 (d) 271
  27. 27.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6404 (c) 6084 (d) 6724
  28. 28.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  29. 29.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 37 (c) 61 (d) 49
  30. 30.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7569 (b) 8649 (c) 8091 (d) 7579
  31. 31.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 141 (c) 121 (d) 101
  32. 32.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 451 (d) 441
  33. 33.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 2 and 4 (c) 1 and 5 (d) 2 and 5
  34. 34.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 324 (c) 4 (d) 170
  35. 35.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  36. 36.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 49 (c) 73 (d) 25
  37. 37.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 1 and 35 (c) 1 and 11 (d) 2 and 10
  38. 38.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 4 and 7 (c) 2 and 14 (d) 1 and 28
  39. 39.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2209 (c) 2509 (d) 2809
  40. 40.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8281 (d) 8291

Answer key

  1. (x + 4)(x + 5)
  2. (x + 3)(x + 9)
  3. 1
  4. 324
  5. 170
  6. 34
  7. (x + 5)(x + 8)
  8. 7
  9. 4489
  10. (x + 3)(x + 6)
  11. (x + 2)(x + 3)
  12. 256
  13. 8 and 9
  14. 9
  15. 81
  16. (x + 2)(x + 4)
  17. (x + 7)(x + 9)
  18. 4
  19. 15
  20. 13
  21. 18
  22. 1225
  23. 64
  24. (x + 2)(x + 9)
  25. (x + 3)(x + 8)
  26. 181
  27. 6724
  28. (x + 6)(x + 7)
  29. 37
  30. 7569
  31. 101
  32. 441
  33. 2 and 4
  34. 16
  35. (x + 6)(x + 8)
  36. 25
  37. 5 and 7
  38. 4 and 7
  39. 2809
  40. 8281

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