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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 686 · 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 + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
  2. 2.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 171 (c) 333 (d) 225
  3. 3.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 1 (d) 271
  4. 4.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 537 (c) 273 (d) 361
  5. 5.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
  6. 6.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 4 (d) 202
  7. 7.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x − 4)(x − 8)
  8. 8.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 72 (d) 16
  9. 9.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 40 (c) 88 (d) 64
  10. 10.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 22 (c) 16 (d) 13
  11. 11.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4884 (c) 4366 (d) 5476
  12. 12.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
  13. 13.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 13 (c) 40 (d) 14
  14. 14.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 156 (c) 116 (d) 276
  15. 15.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 7 (c) 8 (d) 12
  16. 16.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 202 (d) 400
  17. 17.What are the roots of x² − 13x + 40 = 0? (a) 2 and 20 (b) 1 and 40 (c) 1 and 12 (d) 5 and 8
  18. 18.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2926 (d) 2916
  19. 19.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6899 (c) 5929 (d) 6889
  20. 20.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 441 (c) 399 (d) 371
  21. 21.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 15 (c) 57 (d) 56
  22. 22.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9999 (c) 10201 (d) 9811
  23. 23.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
  24. 24.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 14 (d) 16
  25. 25.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 286 (c) 106 (d) 151
  26. 26.If a + b = 12 and ab = 20, what is a² + b²? (a) 104 (b) 124 (c) 184 (d) 144
  27. 27.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
  28. 28.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 1 (d) 113
  29. 29.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 180 (c) 6 (d) 36
  30. 30.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2314 (d) 2304
  31. 31.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 169 (d) 1
  32. 32.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 9 (c) 3 (d) 29
  33. 33.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
  34. 34.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8464 (c) 8474 (d) 8104
  35. 35.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1296 (c) 1936 (d) 1584
  36. 36.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 9 (c) 10 (d) 14
  37. 37.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)
  38. 38.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 916 (c) 676 (d) 1166
  39. 39.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 36 (c) 6 (d) 64
  40. 40.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)

Answer key

  1. (x + 7)(x + 9)
  2. 117
  3. 1
  4. 185
  5. 15
  6. 4
  7. (x + 4)(x + 8)
  8. 17
  9. 40
  10. 10
  11. 4356
  12. (x + 6)(x + 9)
  13. 13
  14. 116
  15. 7
  16. 202
  17. 5 and 8
  18. 2916
  19. 6889
  20. 361
  21. 56
  22. 9801
  23. 181
  24. 15
  25. 106
  26. 104
  27. (x + 4)(x + 9)
  28. 1
  29. 36
  30. 2304
  31. 1
  32. 9
  33. (x + 4)(x + 6)
  34. 8464
  35. 1296
  36. 9
  37. (x + 6)(x + 7)
  38. 1156
  39. 36
  40. (x + 5)(x + 6)

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