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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 572 · 40 questions
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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 = 14 and ab = 48, what is a² + b²? (a) 196 (b) 292 (c) 148 (d) 100
  2. 2.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 5 (c) 157 (d) 25
  3. 3.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 53 (c) 81 (d) 25
  4. 4.If a + b = 11 and ab = 28, what is (a − b)²? (a) 121 (b) 9 (c) 65 (d) 3
  5. 5.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 64 (c) 36 (d) 6
  6. 6.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 73 (d) 17
  7. 7.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 35 (c) 36 (d) 12
  8. 8.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 2 (c) 4 (d) 144
  9. 9.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 16 (c) 2 and 8 (d) 1 and 9
  10. 10.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1936 (c) 1584 (d) 1296
  11. 11.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x − 2)(x − 3)
  12. 12.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 9 (c) 10 (d) 7
  13. 13.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
  14. 14.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 277 (d) 193
  15. 15.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 142 (c) 100 (d) 58
  16. 16.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 28 (c) 26 (d) 12
  17. 17.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10826 (c) 10816 (d) 10016
  18. 18.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 49 (c) 121 (d) 85
  19. 19.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
  20. 20.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8104 (d) 8464
  21. 21.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 1 and 12 (c) 2 and 6 (d) 3 and 4
  22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
  23. 23.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1225 (c) 2025 (d) 1625
  24. 24.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1609 (c) 1859 (d) 1849
  25. 25.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 8 (c) 10 (d) 6
  26. 26.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181
  27. 27.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8291 (d) 8281
  28. 28.What are the roots of x² − 14x + 45 = 0? (a) 3 and 15 (b) 1 and 45 (c) 1 and 13 (d) 5 and 9
  29. 29.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 32 (c) 1 and 11 (d) 4 and 8
  30. 30.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 7225 (c) 5625 (d) 6375
  31. 31.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x − 5)(x − 8) (d) (x + 5)(x + 8)
  32. 32.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
  33. 33.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3731 (c) 3721 (d) 3481
  34. 34.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 324 (c) 36 (d) 180
  35. 35.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
  36. 36.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 9 (d) 11
  37. 37.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2714 (d) 2504
  38. 38.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 225 (c) 125 (d) 5
  39. 39.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  40. 40.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 225 (c) 81 (d) 9

Answer key

  1. 100
  2. 25
  3. 25
  4. 9
  5. 36
  6. 72
  7. 35
  8. 4
  9. 2 and 8
  10. 1296
  11. (x + 2)(x + 3)
  12. 10
  13. 8 and 9
  14. 193
  15. 58
  16. 27
  17. 10816
  18. 49
  19. (x + 2)(x + 6)
  20. 8464
  21. 2 and 6
  22. (x + 6)(x + 7)
  23. 2025
  24. 1849
  25. 7
  26. 181
  27. 8281
  28. 5 and 9
  29. 4 and 8
  30. 5625
  31. (x + 5)(x + 8)
  32. (x + 3)(x + 4)
  33. 3721
  34. 36
  35. (x + 2)(x + 8)
  36. 10
  37. 2704
  38. 25
  39. (x + 7)(x + 8)
  40. 81

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