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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 739 · 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 + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  2. 2.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2516 (c) 2116 (d) 2926
  3. 3.If a + b = 15 and ab = 44, what is a² + b²? (a) 137 (b) 225 (c) 181 (d) 313
  4. 4.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 12 (c) 10 (d) 11
  5. 5.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 6 (c) 1 and 12 (d) 2 and 6
  6. 6.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 9 (c) 225 (d) 441
  7. 7.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  8. 8.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
  9. 9.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 1089 (d) 729
  10. 10.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5786 (c) 5776 (d) 7056
  11. 11.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 324 (d) 164
  12. 12.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
  13. 13.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 34 (c) 2 (d) 4
  14. 14.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 13 (d) 11
  15. 15.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 21 (c) 9 (d) 10
  16. 16.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 1 (c) 181 (d) 271
  17. 17.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 196 (c) 2 (d) 4
  18. 18.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 25 (c) 24 (d) 23
  19. 19.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 57 (d) 56
  20. 20.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 1)(x + 16)
  21. 21.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 73 (c) 97 (d) 81
  22. 22.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
  23. 23.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 11 (c) 10 (d) 9
  24. 24.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 29 (d) 11
  25. 25.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2314 (d) 2304
  26. 26.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 169 (c) 127 (d) 85
  27. 27.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
  28. 28.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 4)(x + 6) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
  29. 29.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 9604 (c) 10004 (d) 10404
  30. 30.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 14 (d) 12
  31. 31.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 12 (c) 35 (d) 36
  32. 32.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 2 (c) 4 (d) 36
  33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
  34. 34.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 5 (c) 6 (d) 7
  35. 35.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 352 (c) 208 (d) 256
  36. 36.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 36 (c) 16 (d) 4
  37. 37.If a + b = 11 and ab = 10, what is a² + b²? (a) 111 (b) 101 (c) 121 (d) 141
  38. 38.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 170 (c) 324 (d) 478
  39. 39.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 15 (c) 16 (d) 54
  40. 40.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3259 (b) 3591 (c) 3969 (d) 3249

Answer key

  1. (x + 8)(x + 9)
  2. 2916
  3. 137
  4. 11
  5. 3 and 4
  6. 9
  7. (x + 6)(x + 7)
  8. (x + 6)(x + 8)
  9. 1089
  10. 5776
  11. 164
  12. (x + 4)(x + 9)
  13. 4
  14. 12
  15. 10
  16. 1
  17. 4
  18. 24
  19. 56
  20. (x + 2)(x + 8)
  21. 65
  22. 17
  23. 10
  24. 30
  25. 2304
  26. 85
  27. (x + 3)(x + 6)
  28. (x + 4)(x + 6)
  29. 10404
  30. 13
  31. 35
  32. 4
  33. (x + 7)(x + 9)
  34. 5
  35. 160
  36. 16
  37. 101
  38. 170
  39. 15
  40. 3249

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