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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 402 · 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 + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 529 (c) 391 (d) 299
  3. 3.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
  4. 4.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
  5. 5.If a + b = 11 and ab = 30, what is a² + b²? (a) 61 (b) 181 (c) 91 (d) 121
  6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
  7. 7.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
  8. 8.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2504 (d) 2714
  9. 9.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 15 (d) 55
  10. 10.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 12 (d) 11
  11. 11.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
  12. 12.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2304 (d) 2314
  13. 13.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 2 and 36 (d) 1 and 16
  14. 14.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 13 (c) 11 (d) 35
  15. 15.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3721 (c) 3731 (d) 3601
  16. 16.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)
  17. 17.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 5 (c) 25 (d) 157
  18. 18.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 36 (c) 146 (d) 6
  19. 19.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 49 (c) 47 (d) 14
  20. 20.If a + b = 9 and ab = 14, what is a² + b²? (a) 67 (b) 81 (c) 109 (d) 53
  21. 21.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 541 (c) 361 (d) 271
  22. 22.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 433 (d) 217
  23. 23.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 4 (c) 144 (d) 74
  24. 24.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
  25. 25.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
  26. 26.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 69 (c) 39 (d) 49
  27. 27.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 8836 (c) 7406 (d) 7396
  28. 28.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6251 (d) 6241
  29. 29.If a + b = 18 and ab = 72, what is (a − b)²? (a) 6 (b) 180 (c) 324 (d) 36
  30. 30.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 4 (c) 2 (d) 400
  31. 31.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
  32. 32.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  33. 33.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
  34. 34.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 2 and 15 (c) 1 and 30 (d) 1 and 10
  35. 35.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 16 (d) 17
  36. 36.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 12 (c) 13 (d) 7
  37. 37.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 5 (c) 9 (d) 7
  38. 38.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 10025 (c) 11035 (d) 11025
  39. 39.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
  40. 40.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 16 (c) 54 (d) 14

Answer key

  1. (x + 3)(x + 4)
  2. 289
  3. (x + 5)(x + 9)
  4. (x + 5)(x + 8)
  5. 61
  6. (x + 8)(x + 9)
  7. (x + 3)(x + 7)
  8. 2704
  9. 56
  10. 11
  11. (x + 3)(x + 5)
  12. 2304
  13. 8 and 9
  14. 12
  15. 3721
  16. (x + 2)(x + 3)
  17. 25
  18. 36
  19. 48
  20. 53
  21. 181
  22. 145
  23. 4
  24. (x + 7)(x + 9)
  25. (x + 2)(x + 7)
  26. 29
  27. 7396
  28. 6241
  29. 36
  30. 4
  31. (x + 2)(x + 4)
  32. (x + 6)(x + 8)
  33. (x + 4)(x + 8)
  34. 5 and 6
  35. 16
  36. 12
  37. 1
  38. 11025
  39. (x + 4)(x + 5)
  40. 15

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