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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 814 · 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 = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 68 (d) 100
  2. 2.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x − 2)(x − 5)
  3. 3.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
  4. 4.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 89 (c) 3 (d) 169
  5. 5.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2714 (d) 2504
  6. 6.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 217 (d) 289
  7. 7.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 15 (d) 56
  8. 8.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 100 (c) 144 (d) 122
  9. 9.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 170 (d) 324
  10. 10.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
  11. 11.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 11 (d) 12
  12. 12.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 28 (c) 29 (d) 11
  13. 13.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3854 (c) 3604 (d) 3844
  14. 14.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 7 (d) 5
  15. 15.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
  16. 16.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)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 9604 (c) 10004 (d) 10414
  18. 18.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
  19. 19.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 961 (d) 971
  20. 20.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
  21. 21.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)
  22. 22.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
  23. 23.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 14 (d) 42
  24. 24.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 10 (c) 11 (d) 21
  25. 25.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 26 (c) 36 (d) 31
  26. 26.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 3)(x + 8) (c) (x + 1)(x + 24) (d) (x + 1)(x + 10)
  27. 27.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164
  28. 28.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6241 (d) 6571
  29. 29.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 277 (c) 193 (d) 529
  30. 30.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 11 (c) 18 (d) 10
  31. 31.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 586 (d) 576
  32. 32.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
  33. 33.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 875 (c) 1225 (d) 635
  34. 34.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 6425 (c) 7235 (d) 7225
  35. 35.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 208 (c) 80 (d) 144
  36. 36.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 21 (c) 33 (d) 17
  37. 37.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3731 (d) 3601
  38. 38.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 40 (c) 16 (d) 4
  39. 39.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
  40. 40.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 45 (d) 14

Answer key

  1. 68
  2. (x + 2)(x + 5)
  3. (x + 5)(x + 9)
  4. 9
  5. 2704
  6. 145
  7. 56
  8. 100
  9. 170
  10. (x + 3)(x + 4)
  11. 12
  12. 28
  13. 3844
  14. 5
  15. (x + 6)(x + 9)
  16. (x + 7)(x + 8)
  17. 10404
  18. (x + 8)(x + 9)
  19. 961
  20. (x + 5)(x + 7)
  21. (x + 4)(x + 6)
  22. (x + 4)(x + 5)
  23. 13
  24. 10
  25. 26
  26. (x + 3)(x + 8)
  27. 164
  28. 6561
  29. 193
  30. 11
  31. 576
  32. 181
  33. 625
  34. 7225
  35. 80
  36. 17
  37. 3721
  38. 16
  39. 17
  40. 14

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