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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 395 · 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 = 18 and ab = 77, what is a² + b²? (a) 170 (b) 324 (c) 247 (d) 478
  2. 2.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 5625 (c) 4875 (d) 4235
  3. 3.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)
  4. 4.What is the product of the roots of x² − 13x + 36 = 0? (a) 35 (b) 37 (c) 36 (d) 13
  5. 5.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 1 and 11 (c) 5 and 7 (d) 2 and 10
  6. 6.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x − 2)(x − 4) (d) (x + 2)(x + 4)
  7. 7.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
  8. 8.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 14 (c) 2 and 27 (d) 1 and 54
  9. 9.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9604 (b) 9614 (c) 9996 (d) 10404
  10. 10.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
  11. 11.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 85 (d) 127
  12. 12.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x − 2)(x − 8)
  13. 13.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4489 (b) 4891 (c) 4499 (d) 5329
  14. 14.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
  15. 15.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
  16. 16.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 1)(x + 8)
  17. 17.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 8 (c) 7 (d) 6
  18. 18.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2035 (c) 1225 (d) 2025
  19. 19.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 15 (d) 56
  20. 20.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 9)
  21. 21.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 149 (c) 289 (d) 219
  22. 22.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 8 (c) 6 (d) 10
  23. 23.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 286 (c) 106 (d) 151
  24. 24.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 7931 (c) 8099 (d) 8281
  25. 25.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 156 (c) 196 (d) 116
  26. 26.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 72 (d) 71
  27. 27.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
  28. 28.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 784 (c) 896 (d) 794
  29. 29.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
  30. 30.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 8 (c) 6 (d) 7
  31. 31.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
  32. 32.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 324 (c) 4 (d) 164
  33. 33.What is the sum of the roots of x² − 8x + 15 = 0? (a) 7 (b) 8 (c) 9 (d) 15
  34. 34.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)
  35. 35.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 76 (c) 52 (d) 100
  36. 36.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)
  37. 37.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 875 (c) 625 (d) 1225
  38. 38.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 175 (c) 225 (d) 325
  39. 39.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 10 (d) 12
  40. 40.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 32 (c) 33 (d) 31

Answer key

  1. 170
  2. 4225
  3. (x + 6)(x + 7)
  4. 36
  5. 5 and 7
  6. (x + 2)(x + 4)
  7. (x + 6)(x + 8)
  8. 6 and 9
  9. 9604
  10. (x + 6)(x + 9)
  11. 85
  12. (x + 2)(x + 8)
  13. 4489
  14. (x + 4)(x + 6)
  15. 8 and 9
  16. (x + 2)(x + 7)
  17. 6
  18. 2025
  19. 15
  20. (x + 3)(x + 7)
  21. 149
  22. 7
  23. 106
  24. 7921
  25. 116
  26. 72
  27. (x + 8)(x + 9)
  28. 784
  29. (x + 7)(x + 9)
  30. 7
  31. (x + 2)(x + 3)
  32. 4
  33. 8
  34. (x + 7)(x + 8)
  35. 52
  36. (x + 5)(x + 8)
  37. 625
  38. 125
  39. 11
  40. 32

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