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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 408 · 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 = 72, what is (a − b)²? (a) 324 (b) 180 (c) 36 (d) 6
  2. 2.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 625 (b) 925 (c) 1235 (d) 1225
  3. 3.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
  4. 4.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)
  5. 5.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 5041 (c) 4761 (d) 4901
  6. 6.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)
  7. 7.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 11 (c) 12 (d) 13
  8. 8.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 10 (c) 24 (d) 9
  9. 9.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 1)(x + 24)
  10. 10.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3721 (c) 3599 (d) 3481
  11. 11.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)
  12. 12.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 225 (c) 113 (d) 169
  13. 13.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 43 (d) 13
  14. 14.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 1)(x + 9) (d) (x + 2)(x + 8)
  15. 15.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3374 (c) 3844 (d) 3364
  16. 16.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 2 (c) 4 (d) 130
  17. 17.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
  18. 18.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2314 (c) 2304 (d) 2496
  19. 19.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 916 (c) 676 (d) 1156
  20. 20.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2304 (c) 2504 (d) 2704
  21. 21.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)
  22. 22.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 9 (c) 289 (d) 149
  23. 23.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 1 and 30 (c) 2 and 15 (d) 5 and 6
  24. 24.If a + b = 15 and ab = 54, what is a² + b²? (a) 117 (b) 225 (c) 333 (d) 171
  25. 25.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 1)(x + 7)
  26. 26.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 21 (c) 20 (d) 22
  27. 27.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6724 (c) 6084 (d) 6396
  28. 28.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 361 (d) 185
  29. 29.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 1 and 56 (c) 2 and 28 (d) 7 and 8
  30. 30.If a + b = 11 and ab = 24, what is (a − b)²? (a) 25 (b) 121 (c) 73 (d) 5
  31. 31.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 124 (c) 104 (d) 144
  32. 32.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 145 (c) 289 (d) 1
  33. 33.If a + b = 7 and ab = 6, what is a² + b²? (a) 43 (b) 61 (c) 37 (d) 49
  34. 34.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 409 (c) 289 (d) 229
  35. 35.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 111 (c) 121 (d) 101
  36. 36.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
  37. 37.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 29 (c) 31 (d) 30
  38. 38.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 72 (d) 17
  39. 39.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
  40. 40.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 50 (c) 36 (d) 6

Answer key

  1. 36
  2. 1225
  3. (x + 6)(x + 7)
  4. (x + 5)(x + 8)
  5. 5041
  6. (x + 6)(x + 9)
  7. 12
  8. 10
  9. (x + 4)(x + 6)
  10. 3481
  11. (x + 5)(x + 6)
  12. 113
  13. 42
  14. (x + 2)(x + 8)
  15. 3364
  16. 4
  17. (x + 3)(x + 5)
  18. 2304
  19. 1156
  20. 2704
  21. (x + 2)(x + 3)
  22. 9
  23. 5 and 6
  24. 117
  25. (x + 2)(x + 6)
  26. 21
  27. 6084
  28. 185
  29. 7 and 8
  30. 25
  31. 104
  32. 1
  33. 37
  34. 169
  35. 101
  36. (x + 2)(x + 9)
  37. 30
  38. 72
  39. 8 and 9
  40. 36

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