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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 716 · 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.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 2)(x + 3)
  2. 2.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3374 (c) 3364 (d) 3596
  3. 3.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 361 (c) 541 (d) 181
  4. 4.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 9801 (c) 10211 (d) 10201
  6. 6.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 13 (c) 35 (d) 36
  7. 7.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 9025 (c) 8125 (d) 9035
  8. 8.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 244 (c) 164 (d) 484
  9. 9.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
  10. 10.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 9 (c) 65 (d) 121
  11. 11.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 304 (d) 400
  12. 12.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9975 (c) 9025 (d) 9035
  13. 13.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 57 (c) 78 (d) 50
  14. 14.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 16 (c) 64 (d) 62
  15. 15.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 8 (c) 14 (d) 15
  16. 16.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)
  17. 17.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 7921 (c) 8101 (d) 8281
  18. 18.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 100 (c) 79 (d) 58
  19. 19.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1849 (c) 1609 (d) 1859
  20. 20.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
  21. 21.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 225 (d) 169
  22. 22.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9999 (c) 9801 (d) 10201
  23. 23.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 676 (c) 1156 (d) 916
  24. 24.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 121 (c) 5 (d) 73
  25. 25.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2304 (c) 2504 (d) 2704
  26. 26.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 49 (c) 73 (d) 37
  27. 27.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 382 (c) 193 (d) 256
  28. 28.If a + b = 12 and ab = 20, what is a² + b²? (a) 104 (b) 184 (c) 124 (d) 144
  29. 29.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 139 (c) 169 (d) 109
  30. 30.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 33 (c) 17 (d) 25
  31. 31.If a + b = 9 and ab = 18, what is a² + b²? (a) 117 (b) 45 (c) 81 (d) 63
  32. 32.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
  33. 33.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6391 (c) 5929 (d) 6889
  34. 34.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
  35. 35.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 85 (c) 103 (d) 121
  36. 36.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 52 (c) 148 (d) 100
  37. 37.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 4 (c) 324 (d) 164
  38. 38.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1774 (c) 1604 (d) 1764
  39. 39.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 8 (c) 6 (d) 7
  40. 40.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 208 (c) 80 (d) 144

Answer key

  1. (x + 2)(x + 3)
  2. 3364
  3. 181
  4. (x + 4)(x + 8)
  5. 10201
  6. 36
  7. 9025
  8. 164
  9. (x + 7)(x + 8)
  10. 9
  11. 208
  12. 9025
  13. 50
  14. 63
  15. 15
  16. (x + 6)(x + 8)
  17. 8281
  18. 58
  19. 1849
  20. (x + 3)(x + 8)
  21. 1
  22. 9801
  23. 1156
  24. 25
  25. 2704
  26. 25
  27. 130
  28. 104
  29. 109
  30. 17
  31. 45
  32. (x + 5)(x + 8)
  33. 5929
  34. (x + 6)(x + 9)
  35. 85
  36. 52
  37. 4
  38. 1764
  39. 7
  40. 80

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