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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 708 · 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 = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 400 (c) 202 (d) 4
  2. 2.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 247 (c) 170 (d) 478
  3. 3.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 7569 (c) 8109 (d) 8649
  4. 4.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
  5. 5.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 289 (c) 145 (d) 433
  6. 6.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 14 (c) 12 (d) 13
  7. 7.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 8 (c) 7 (d) 6
  8. 8.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
  9. 9.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 49 (c) 29 (d) 69
  10. 10.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
  11. 11.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  12. 12.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
  13. 13.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10211 (c) 9801 (d) 10201
  14. 14.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 9 (d) 149
  15. 15.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 49 (d) 47
  16. 16.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 271 (d) 361
  17. 17.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2501 (c) 2401 (d) 2601
  18. 18.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 85 (c) 169 (d) 1
  19. 19.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6561 (d) 6571
  20. 20.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 36 (c) 4 (d) 2
  21. 21.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 144 (c) 117 (d) 198
  22. 22.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 15 (c) 16 (d) 63
  23. 23.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 147 (c) 125 (d) 169
  24. 24.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 1024 (c) 794 (d) 784
  25. 25.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 7921 (c) 8101 (d) 8281
  26. 26.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 3 and 5 (c) 1 and 15 (d) 2 and 6
  27. 27.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)
  28. 28.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 202 (d) 598
  29. 29.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 396 (c) 484 (d) 334
  30. 30.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
  31. 31.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 11 (c) 4 and 8 (d) 1 and 32
  32. 32.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3609 (c) 3969 (d) 3979
  33. 33.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)
  34. 34.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 12 (d) 11
  35. 35.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 10) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
  36. 36.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 73 (d) 71
  37. 37.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 4 (c) 16 (d) 58
  38. 38.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 11 (c) 12 (d) 27
  39. 39.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 7 (d) 5
  40. 40.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 10 (c) 24 (d) 9

Answer key

  1. 4
  2. 170
  3. 8649
  4. (x + 8)(x + 9)
  5. 145
  6. 13
  7. 6
  8. 17
  9. 29
  10. (x + 2)(x + 8)
  11. (x + 7)(x + 8)
  12. (x + 5)(x + 8)
  13. 10201
  14. 9
  15. 48
  16. 181
  17. 2601
  18. 1
  19. 6561
  20. 4
  21. 90
  22. 16
  23. 125
  24. 784
  25. 8281
  26. 3 and 5
  27. (x + 3)(x + 9)
  28. 202
  29. 324
  30. (x + 7)(x + 9)
  31. 4 and 8
  32. 3969
  33. (x + 4)(x + 6)
  34. 11
  35. (x + 4)(x + 7)
  36. 72
  37. 16
  38. 12
  39. 5
  40. 10

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