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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 643 · 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 + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
  2. 2.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 111 (c) 101 (d) 121
  3. 3.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  4. 4.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 4 (d) 400
  5. 5.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 81 (c) 61 (d) 1
  6. 6.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
  7. 7.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 12 (d) 11
  8. 8.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 172 (c) 196 (d) 148
  9. 9.If a + b = 12 and ab = 11, what is a² + b²? (a) 133 (b) 122 (c) 144 (d) 166
  10. 10.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 62 (c) 16 (d) 64
  11. 11.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
  12. 12.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 4 (c) 2 (d) 100
  13. 13.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 71 (c) 17 (d) 73
  14. 14.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 4624 (c) 5194 (d) 5184
  15. 15.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 144 (c) 198 (d) 117
  16. 16.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 2)(x + 7) (c) (x + 1)(x + 14) (d) (x − 2)(x − 7)
  17. 17.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  18. 18.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
  19. 19.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
  20. 20.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)
  21. 21.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1849 (d) 1609
  22. 22.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 256 (c) 196 (d) 136
  23. 23.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 52 (c) 28 (d) 36
  24. 24.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 26 (c) 28 (d) 27
  25. 25.If a + b = 14 and ab = 40, what is (a − b)²? (a) 36 (b) 196 (c) 6 (d) 116
  26. 26.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2601 (c) 2501 (d) 2401
  27. 27.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3979 (c) 3249 (d) 3969
  28. 28.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 81 (c) 9 (d) 3
  29. 29.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  30. 30.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 10 (c) 11 (d) 28
  31. 31.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
  32. 32.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
  33. 33.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)
  34. 34.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 79 (c) 100 (d) 142
  35. 35.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 157 (c) 421 (d) 289
  36. 36.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 81 (c) 63 (d) 117
  37. 37.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 225 (c) 657 (d) 333
  38. 38.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
  39. 39.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 6 and 7 (d) 2 and 21
  40. 40.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 23 (c) 11 (d) 24

Answer key

  1. (x + 6)(x + 7)
  2. 101
  3. (x + 6)(x + 8)
  4. 4
  5. 1
  6. (x + 3)(x + 8)
  7. 12
  8. 148
  9. 122
  10. 63
  11. (x + 3)(x + 7)
  12. 4
  13. 72
  14. 5184
  15. 90
  16. (x + 2)(x + 7)
  17. 17
  18. (x + 2)(x + 8)
  19. 56
  20. (x + 2)(x + 6)
  21. 1849
  22. 136
  23. 20
  24. 27
  25. 36
  26. 2601
  27. 3969
  28. 9
  29. (x + 7)(x + 8)
  30. 11
  31. (x + 4)(x + 5)
  32. (x + 2)(x + 9)
  33. (x + 5)(x + 9)
  34. 58
  35. 157
  36. 45
  37. 225
  38. (x + 5)(x + 7)
  39. 6 and 7
  40. 24

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