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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 697 · 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 99²? (a) 9999 (b) 9811 (c) 9801 (d) 10201
  2. 2.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5476 (d) 5486
  3. 3.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 5 (c) 121 (d) 73
  4. 4.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1609 (c) 1849 (d) 1859
  5. 5.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 63 (c) 16 (d) 64
  6. 6.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 416 (c) 586 (d) 256
  7. 7.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
  8. 8.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
  9. 9.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 27 (c) 11 (d) 12
  10. 10.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  11. 11.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6251 (c) 6399 (d) 6241
  12. 12.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 1)(x + 28)
  13. 13.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 45 (c) 117 (d) 81
  14. 14.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 784 (c) 794 (d) 1024
  15. 15.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 4 (c) 1 and 6 (d) 2 and 4
  16. 16.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
  17. 17.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 5476 (c) 4884 (d) 4366
  18. 18.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
  19. 19.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 11 (c) 35 (d) 12
  20. 20.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
  21. 21.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 81 (c) 9 (d) 45
  22. 22.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 12 (d) 13
  23. 23.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 10 (c) 21 (d) 9
  24. 24.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)
  25. 25.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 74 (c) 4 (d) 144
  26. 26.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 97 (c) 169 (d) 73
  27. 27.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)
  28. 28.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 10201 (c) 9801 (d) 10211
  29. 29.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 145 (d) 217
  30. 30.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 10) (c) (x + 1)(x + 6) (d) (x − 2)(x − 5)
  31. 31.What is the product of the roots of x² − 12x + 35 = 0? (a) 12 (b) 36 (c) 34 (d) 35
  32. 32.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2209 (c) 2509 (d) 2819
  33. 33.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2601 (c) 2401 (d) 2499
  34. 34.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)
  35. 35.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 304 (c) 208 (d) 400
  36. 36.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 4 (d) 5
  37. 37.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4909 (c) 5329 (d) 4489
  38. 38.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 1156 (c) 686 (d) 884
  39. 39.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 441 (c) 361 (d) 401
  40. 40.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3591 (c) 3249 (d) 3259

Answer key

  1. 9801
  2. 5476
  3. 25
  4. 1849
  5. 63
  6. 576
  7. (x + 4)(x + 8)
  8. (x + 6)(x + 8)
  9. 12
  10. (x + 5)(x + 6)
  11. 6241
  12. (x + 4)(x + 7)
  13. 45
  14. 784
  15. 2 and 3
  16. (x + 6)(x + 7)
  17. 4356
  18. (x + 6)(x + 9)
  19. 12
  20. 17
  21. 9
  22. 13
  23. 10
  24. (x + 2)(x + 4)
  25. 4
  26. 73
  27. (x + 7)(x + 9)
  28. 10201
  29. 145
  30. (x + 2)(x + 5)
  31. 35
  32. 2809
  33. 2401
  34. (x + 2)(x + 3)
  35. 208
  36. 5
  37. 5329
  38. 676
  39. 441
  40. 3249

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