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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 997 · 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.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 4 (c) 256 (d) 2
  2. 2.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 289 (d) 1
  3. 3.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 8 (c) 196 (d) 130
  4. 4.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 6 (c) 2 and 5 (d) 1 and 10
  5. 5.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
  6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
  7. 7.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 180 (c) 6 (d) 36
  8. 8.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 12 (c) 10 (d) 11
  9. 9.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 484 (c) 334 (d) 396
  10. 10.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 157 (c) 289 (d) 421
  11. 11.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 12 (d) 14
  12. 12.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 225 (d) 333
  13. 13.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 478 (c) 247 (d) 170
  14. 14.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 81 (c) 3 (d) 45
  15. 15.If a + b = 11 and ab = 30, what is (a − b)²? (a) 1 (b) 121 (c) 91 (d) 61
  16. 16.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 71 (d) 73
  17. 17.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
  18. 18.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
  19. 19.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 5 (c) 7 (d) 8
  20. 20.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 4925 (c) 5625 (d) 5635
  21. 21.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
  22. 22.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2401 (c) 2611 (d) 2601
  23. 23.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  24. 24.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 1 and 12 (c) 2 and 21 (d) 6 and 7
  25. 25.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 289 (c) 219 (d) 429
  26. 26.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 9 (c) 361 (d) 3
  27. 27.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 441 (c) 451 (d) 361
  28. 28.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
  29. 29.If a + b = 12 and ab = 20, what is a² + b²? (a) 104 (b) 184 (c) 144 (d) 124
  30. 30.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 56 (c) 1 and 14 (d) 7 and 8
  31. 31.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
  32. 32.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 8 (c) 64 (d) 144
  33. 33.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 19 (c) 25 (d) 1
  34. 34.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 12 (d) 13
  35. 35.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 35 (d) 11
  36. 36.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 85 (c) 169 (d) 127
  37. 37.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 5)(x + 7) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
  38. 38.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
  39. 39.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
  40. 40.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 33 (c) 25 (d) 21

Answer key

  1. 4
  2. 1
  3. 64
  4. 2 and 5
  5. (x + 2)(x + 3)
  6. (x + 8)(x + 9)
  7. 36
  8. 11
  9. 324
  10. 157
  11. 13
  12. 225
  13. 170
  14. 9
  15. 1
  16. 72
  17. (x + 2)(x + 6)
  18. (x + 4)(x + 9)
  19. 6
  20. 5625
  21. (x + 6)(x + 8)
  22. 2601
  23. (x + 7)(x + 9)
  24. 6 and 7
  25. 149
  26. 9
  27. 441
  28. (x + 5)(x + 6)
  29. 104
  30. 7 and 8
  31. (x + 3)(x + 5)
  32. 64
  33. 1
  34. 13
  35. 12
  36. 1
  37. (x + 5)(x + 7)
  38. (x + 2)(x + 5)
  39. (x + 3)(x + 8)
  40. 17

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