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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 842 · 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 + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
  2. 2.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 244 (c) 172 (d) 148
  3. 3.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 16 (d) 63
  4. 4.What is the product of the roots of x² − 12x + 35 = 0? (a) 35 (b) 12 (c) 36 (d) 34
  5. 5.What is the product of the roots of x² − 7x + 12 = 0? (a) 13 (b) 12 (c) 7 (d) 11
  6. 6.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 40 (c) 13 (d) 14
  7. 7.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 21 (c) 10 (d) 22
  8. 8.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 11 (c) 1 and 27 (d) 3 and 9
  9. 9.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
  10. 10.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
  11. 11.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1296 (c) 1616 (d) 1936
  12. 12.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 8) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
  13. 13.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 12 (c) 6 and 7 (d) 1 and 42
  14. 14.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 6425 (d) 7225
  15. 15.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x − 2)(x − 7) (c) (x + 1)(x + 14) (d) (x + 1)(x + 8)
  16. 16.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 3 and 21 (d) 7 and 9
  17. 17.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 271 (d) 541
  18. 18.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 14 (c) 13 (d) 48
  19. 19.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4366 (c) 5476 (d) 4356
  20. 20.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
  21. 21.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 6 (c) 180 (d) 36
  22. 22.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x − 2)(x − 8) (c) (x + 1)(x + 16) (d) (x + 2)(x + 8)
  23. 23.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 9 (c) 117 (d) 225
  24. 24.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 11 (c) 28 (d) 10
  25. 25.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
  26. 26.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 7) (d) (x + 1)(x + 12)
  27. 27.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 421 (c) 289 (d) 223
  28. 28.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)
  29. 29.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 10 (c) 8 (d) 9
  30. 30.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 151 (c) 286 (d) 106
  31. 31.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 64 (c) 196 (d) 8
  32. 32.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  33. 33.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 1 and 14 (d) 7 and 8
  34. 34.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
  35. 35.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6561 (c) 6399 (d) 6241
  36. 36.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 13 (c) 27 (d) 12
  37. 37.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 1089 (d) 729
  38. 38.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2401 (c) 2601 (d) 2499
  39. 39.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 36 (c) 100 (d) 68
  40. 40.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271

Answer key

  1. (x + 4)(x + 6)
  2. 148
  3. 16
  4. 35
  5. 12
  6. 13
  7. 21
  8. 3 and 9
  9. 17
  10. (x + 3)(x + 5)
  11. 1936
  12. (x + 2)(x + 4)
  13. 6 and 7
  14. 7225
  15. (x + 2)(x + 7)
  16. 7 and 9
  17. 181
  18. 14
  19. 4356
  20. (x + 8)(x + 9)
  21. 36
  22. (x + 2)(x + 8)
  23. 9
  24. 11
  25. (x + 5)(x + 7)
  26. (x + 2)(x + 6)
  27. 157
  28. (x + 5)(x + 9)
  29. 9
  30. 106
  31. 64
  32. (x + 5)(x + 8)
  33. 7 and 8
  34. (x + 6)(x + 8)
  35. 6241
  36. 12
  37. 1089
  38. 2401
  39. 36
  40. 1

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