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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 278 · 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.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 27 (c) 26 (d) 12
  2. 2.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
  3. 3.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x − 2)(x − 9) (c) (x + 1)(x + 10) (d) (x + 1)(x + 18)
  4. 4.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1521 (c) 1531 (d) 1599
  5. 5.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 94 (c) 64 (d) 49
  6. 6.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 81 (c) 65 (d) 73
  7. 7.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 2 (c) 202 (d) 400
  8. 8.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 276 (c) 156 (d) 116
  9. 9.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 9 (c) 7 (d) 5
  10. 10.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
  11. 11.If a + b = 14 and ab = 45, what is (a − b)²? (a) 4 (b) 106 (c) 196 (d) 16
  12. 12.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 2 and 36 (d) 1 and 72
  13. 13.If a + b = 15 and ab = 36, what is (a − b)²? (a) 153 (b) 9 (c) 225 (d) 81
  14. 14.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1596 (c) 1444 (d) 1454
  15. 15.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 100 (c) 292 (d) 148
  16. 16.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 17 (d) 71
  17. 17.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  18. 18.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 4 (c) 164 (d) 2
  19. 19.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2601 (d) 2401
  20. 20.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 13 (c) 36 (d) 14
  21. 21.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
  22. 22.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 9811 (c) 10201 (d) 9801
  23. 23.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 169 (c) 253 (d) 85
  24. 24.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  25. 25.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 625 (c) 235 (d) 225
  26. 26.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 1 and 12 (d) 6 and 7
  27. 27.What is the product of the roots of x² − 14x + 45 = 0? (a) 45 (b) 46 (c) 44 (d) 14
  28. 28.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 289 (c) 25 (d) 5
  29. 29.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
  30. 30.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 635 (c) 225 (d) 425
  31. 31.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 6375 (c) 5635 (d) 5625
  32. 32.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 1 and 6 (c) 3 and 4 (d) 1 and 12
  33. 33.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 14 (c) 12 (d) 40
  34. 34.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13
  35. 35.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7931 (c) 8099 (d) 7921
  36. 36.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 1 and 5 (c) 1 and 8 (d) 2 and 5
  37. 37.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)
  38. 38.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 5 and 6 (c) 1 and 30 (d) 2 and 15
  39. 39.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  40. 40.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 2)(x + 5) (c) (x − 2)(x − 5) (d) (x + 1)(x + 6)

Answer key

  1. 27
  2. (x + 5)(x + 7)
  3. (x + 2)(x + 9)
  4. 1521
  5. 34
  6. 65
  7. 4
  8. 116
  9. 5
  10. (x + 6)(x + 7)
  11. 16
  12. 8 and 9
  13. 81
  14. 1444
  15. 100
  16. 72
  17. (x + 7)(x + 8)
  18. 4
  19. 2601
  20. 13
  21. 17
  22. 9801
  23. 85
  24. (x + 4)(x + 8)
  25. 225
  26. 6 and 7
  27. 45
  28. 25
  29. (x + 3)(x + 9)
  30. 625
  31. 5625
  32. 3 and 4
  33. 13
  34. 14
  35. 7921
  36. 2 and 4
  37. (x + 4)(x + 6)
  38. 5 and 6
  39. (x + 8)(x + 9)
  40. (x + 2)(x + 5)

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