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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 904 · 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 sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 32 (c) 13 (d) 12
  2. 2.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3374 (c) 3844 (d) 3364
  3. 3.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 8464 (c) 7744 (d) 7754
  4. 4.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 144 (c) 6 (d) 36
  5. 5.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)
  6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
  7. 7.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 5 (c) 25 (d) 361
  8. 8.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 9 (c) 1 (d) 5
  9. 9.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 441 (c) 401 (d) 361
  10. 10.If a + b = 8 and ab = 15, what is (a − b)²? (a) 4 (b) 34 (c) 2 (d) 64
  11. 11.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
  12. 12.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)
  13. 13.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2809 (c) 2219 (d) 2209
  14. 14.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 592 (c) 304 (d) 208
  15. 15.If a + b = 11 and ab = 24, what is (a − b)²? (a) 5 (b) 121 (c) 73 (d) 25
  16. 16.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 7225 (d) 6375
  17. 17.What are the roots of x² − 10x + 16 = 0? (a) 1 and 16 (b) 2 and 8 (c) 1 and 9 (d) 3 and 7
  18. 18.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 225 (c) 5 (d) 25
  19. 19.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 100 (c) 76 (d) 52
  20. 20.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 1 (c) 41 (d) 61
  21. 21.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 2 (c) 16 (d) 4
  22. 22.If a + b = 11 and ab = 18, what is a² + b²? (a) 85 (b) 157 (c) 103 (d) 121
  23. 23.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
  24. 24.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
  25. 25.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 391 (c) 299 (d) 289
  26. 26.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 80 (c) 208 (d) 144
  27. 27.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 4 (c) 196 (d) 2
  28. 28.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 202 (d) 400
  29. 29.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
  30. 30.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)
  31. 31.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 1 (d) 289
  32. 32.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3844 (c) 3854 (d) 3604
  33. 33.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10009 (c) 10609 (d) 10619
  34. 34.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)
  35. 35.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x − 4)(x − 6)
  36. 36.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 10201 (c) 9999 (d) 9801
  37. 37.What are the roots of x² − 10x + 24 = 0? (a) 4 and 6 (b) 1 and 9 (c) 1 and 24 (d) 2 and 12
  38. 38.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 30 (c) 11 (d) 31
  39. 39.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 3 (c) 121 (d) 9
  40. 40.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)

Answer key

  1. 12
  2. 3364
  3. 7744
  4. 36
  5. (x + 5)(x + 9)
  6. (x + 7)(x + 8)
  7. 25
  8. 1
  9. 441
  10. 4
  11. (x + 5)(x + 8)
  12. (x + 2)(x + 3)
  13. 2209
  14. 208
  15. 25
  16. 5625
  17. 2 and 8
  18. 25
  19. 52
  20. 1
  21. 4
  22. 85
  23. (x + 2)(x + 9)
  24. 17
  25. 289
  26. 80
  27. 4
  28. 4
  29. (x + 3)(x + 4)
  30. (x + 6)(x + 9)
  31. 1
  32. 3844
  33. 10609
  34. (x + 8)(x + 9)
  35. (x + 4)(x + 6)
  36. 9801
  37. 4 and 6
  38. 30
  39. 9
  40. (x + 5)(x + 7)

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