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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 773 · 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 53²? (a) 2809 (b) 2819 (c) 2209 (d) 2509
  2. 2.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 1 and 6 (c) 3 and 4 (d) 2 and 5
  3. 3.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 6416 (c) 7056 (d) 5776
  4. 4.If a + b = 9 and ab = 8, what is a² + b²? (a) 73 (b) 81 (c) 97 (d) 65
  5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
  6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
  7. 7.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 11 (c) 10 (d) 7
  8. 8.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2304 (c) 2704 (d) 2504
  9. 9.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
  10. 10.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 62 (c) 16 (d) 64
  11. 11.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
  12. 12.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 189 (c) 153 (d) 225
  13. 13.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 2)(x + 7) (c) (x − 2)(x − 7) (d) (x + 1)(x + 14)
  14. 14.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 45 (d) 14
  15. 15.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 11 (c) 32 (d) 12
  16. 16.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
  17. 17.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 1 and 9 (d) 4 and 6
  18. 18.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)
  19. 19.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 85 (d) 127
  20. 20.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 256 (c) 160 (d) 352
  21. 21.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5184 (c) 5194 (d) 4624
  22. 22.If a + b = 11 and ab = 18, what is a² + b²? (a) 85 (b) 121 (c) 157 (d) 103
  23. 23.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 45 (c) 3 (d) 9
  24. 24.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 451 (c) 361 (d) 441
  25. 25.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3136 (c) 3584 (d) 4096
  26. 26.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
  27. 27.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)
  28. 28.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 53 (d) 54
  29. 29.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 400 (c) 598 (d) 301
  30. 30.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 223 (c) 421 (d) 289
  31. 31.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1596 (c) 1454 (d) 1764
  32. 32.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 45) (c) (x + 1)(x + 13) (d) (x + 5)(x + 9)
  33. 33.What is the product of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 31 (d) 33
  34. 34.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 2 and 3 (c) 1 and 4 (d) 1 and 6
  35. 35.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 15 (d) 56
  36. 36.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 21 (c) 22 (d) 10
  37. 37.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 3 (c) 225 (d) 441
  38. 38.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 61 (c) 43 (d) 37
  39. 39.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 18) (c) (x + 1)(x + 10) (d) (x − 2)(x − 9)
  40. 40.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 100 (d) 68

Answer key

  1. 2809
  2. 2 and 5
  3. 7056
  4. 65
  5. (x + 8)(x + 9)
  6. (x + 7)(x + 8)
  7. 10
  8. 2704
  9. (x + 4)(x + 6)
  10. 63
  11. (x + 3)(x + 9)
  12. 153
  13. (x + 2)(x + 7)
  14. 14
  15. 12
  16. (x + 6)(x + 8)
  17. 4 and 6
  18. (x + 5)(x + 8)
  19. 85
  20. 160
  21. 5184
  22. 85
  23. 9
  24. 441
  25. 3136
  26. (x + 2)(x + 3)
  27. (x + 5)(x + 7)
  28. 54
  29. 202
  30. 157
  31. 1444
  32. (x + 5)(x + 9)
  33. 32
  34. 2 and 3
  35. 15
  36. 21
  37. 9
  38. 37
  39. (x + 2)(x + 9)
  40. 68

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