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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 301 · 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.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3025 (c) 2525 (d) 3035
  2. 2.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 37 (c) 25 (d) 73
  3. 3.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 11 (c) 13 (d) 12
  4. 4.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 54 (c) 53 (d) 15
  5. 5.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 8281 (c) 8099 (d) 7921
  6. 6.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
  7. 7.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 4 (c) 58 (d) 16
  8. 8.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 361 (d) 271
  9. 9.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 11 (c) 13 (d) 12
  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.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10609 (d) 10619
  12. 12.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 11 (c) 21 (d) 9
  13. 13.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 56 (d) 55
  14. 14.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1609 (c) 1369 (d) 1859
  15. 15.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6401 (d) 6241
  16. 16.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 256 (c) 201 (d) 366
  17. 17.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1936 (c) 1296 (d) 1616
  18. 18.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
  19. 19.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 371 (c) 441 (d) 399
  20. 20.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
  21. 21.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 11 (c) 12 (d) 30
  22. 22.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 4489 (c) 5329 (d) 5339
  23. 23.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
  24. 24.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 686 (c) 884 (d) 676
  25. 25.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 289 (d) 145
  26. 26.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)
  27. 27.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2116 (b) 2916 (c) 2926 (d) 2516
  28. 28.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 18 (d) 72
  29. 29.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3609 (c) 3969 (d) 3249
  30. 30.If a + b = 9 and ab = 14, what is (a − b)²? (a) 5 (b) 25 (c) 81 (d) 53
  31. 31.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 130 (c) 8 (d) 196
  32. 32.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 1)(x + 4)
  33. 33.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 15 (d) 13
  34. 34.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 9 (c) 7 (d) 8
  35. 35.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 4896 (c) 5184 (d) 4634
  36. 36.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 325 (c) 225 (d) 175
  37. 37.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
  38. 38.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 6 (c) 7 (d) 8
  39. 39.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 16 (c) 2 (d) 4
  40. 40.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72

Answer key

  1. 3025
  2. 25
  3. 12
  4. 54
  5. 7921
  6. (x + 2)(x + 7)
  7. 16
  8. 1
  9. 12
  10. (x + 6)(x + 7)
  11. 10609
  12. 10
  13. 56
  14. 1849
  15. 6561
  16. 146
  17. 1936
  18. (x + 6)(x + 8)
  19. 361
  20. (x + 3)(x + 6)
  21. 11
  22. 5329
  23. (x + 5)(x + 6)
  24. 676
  25. 145
  26. (x + 8)(x + 9)
  27. 2916
  28. 17
  29. 3969
  30. 25
  31. 64
  32. (x + 2)(x + 3)
  33. 14
  34. 8
  35. 4624
  36. 125
  37. (x + 4)(x + 7)
  38. 7
  39. 4
  40. 72

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