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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 88 · 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 + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  2. 2.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
  3. 3.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 13 (c) 11 (d) 27
  4. 4.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3146 (c) 3136 (d) 4096
  5. 5.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
  6. 6.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 13 (d) 42
  7. 7.If a + b = 11 and ab = 10, what is (a − b)²? (a) 121 (b) 81 (c) 101 (d) 9
  8. 8.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
  9. 9.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 6384 (c) 7056 (d) 5776
  10. 10.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 352 (d) 160
  11. 11.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 163 (c) 196 (d) 130
  12. 12.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
  13. 13.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
  14. 14.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 169 (c) 97 (d) 73
  15. 15.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 71 (d) 73
  16. 16.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 2 and 6 (c) 1 and 7 (d) 3 and 4
  17. 17.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 5 (c) 289 (d) 25
  18. 18.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 196 (c) 148 (d) 100
  19. 19.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
  20. 20.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1099 (d) 1089
  21. 21.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 13 (d) 48
  22. 22.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 170 (d) 324
  23. 23.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 169 (c) 49 (d) 7
  24. 24.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 11 (c) 10 (d) 7
  25. 25.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2304 (c) 2314 (d) 2496
  26. 26.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 635 (c) 225 (d) 625
  27. 27.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 4 (c) 208 (d) 16
  28. 28.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
  29. 29.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 1 and 10 (c) 2 and 12 (d) 1 and 24
  30. 30.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3491 (c) 3481 (d) 3721
  31. 31.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 219 (c) 429 (d) 289
  32. 32.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5476 (d) 5486
  33. 33.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 6 (c) 12 (d) 7
  34. 34.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 217 (d) 1
  35. 35.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2209 (c) 2809 (d) 2491
  36. 36.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)
  37. 37.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1774 (c) 1764 (d) 1604
  38. 38.What are the roots of x² − 13x + 36 = 0? (a) 4 and 9 (b) 1 and 12 (c) 2 and 18 (d) 1 and 36
  39. 39.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 229 (c) 289 (d) 169
  40. 40.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)

Answer key

  1. (x + 4)(x + 8)
  2. (x + 4)(x + 5)
  3. 12
  4. 3136
  5. (x + 6)(x + 9)
  6. 42
  7. 81
  8. (x + 3)(x + 6)
  9. 5776
  10. 160
  11. 130
  12. (x + 4)(x + 6)
  13. (x + 2)(x + 3)
  14. 73
  15. 72
  16. 2 and 6
  17. 25
  18. 100
  19. 17
  20. 1089
  21. 14
  22. 170
  23. 49
  24. 10
  25. 2304
  26. 625
  27. 16
  28. (x + 3)(x + 5)
  29. 3 and 8
  30. 3481
  31. 149
  32. 5476
  33. 7
  34. 1
  35. 2209
  36. (x + 2)(x + 6)
  37. 1764
  38. 4 and 9
  39. 169
  40. (x + 6)(x + 7)

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