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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 19 · 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 = 10 and ab = 24, what is a² + b²? (a) 100 (b) 52 (c) 76 (d) 148
  2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
  3. 3.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 41 (c) 61 (d) 81
  4. 4.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1454 (c) 1596 (d) 1444
  5. 5.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 289 (c) 223 (d) 157
  6. 6.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
  7. 7.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 44 (c) 45 (d) 46
  8. 8.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 1 (d) 217
  9. 9.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2819 (c) 2209 (d) 2509
  10. 10.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 14 (c) 49 (d) 48
  11. 11.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
  12. 12.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 146 (c) 201 (d) 256
  13. 13.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 9604 (c) 9996 (d) 10404
  14. 14.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 2 and 15 (c) 1 and 10 (d) 1 and 30
  15. 15.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 16 (c) 10 (d) 22
  16. 16.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6396 (c) 6724 (d) 6084
  17. 17.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 5 (c) 8 (d) 6
  18. 18.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 10 (c) 1 and 6 (d) 2 and 5
  19. 19.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x − 2)(x − 6) (c) (x + 1)(x + 12) (d) (x + 1)(x + 7)
  20. 20.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1849 (d) 1609
  21. 21.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)
  22. 22.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 598 (d) 202
  23. 23.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 17 (d) 73
  24. 24.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 2 (d) 400
  25. 25.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
  26. 26.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
  27. 27.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 1 and 16 (c) 2 and 8 (d) 3 and 7
  28. 28.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 104 (c) 144 (d) 124
  29. 29.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 2 and 14 (c) 1 and 10 (d) 1 and 28
  30. 30.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 97 (c) 25 (d) 5
  31. 31.If a + b = 14 and ab = 24, what is (a − b)²? (a) 10 (b) 148 (c) 196 (d) 100
  32. 32.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 24 (c) 1 and 10 (d) 3 and 8
  33. 33.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
  34. 34.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
  35. 35.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 37 (c) 73 (d) 49
  36. 36.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 4 (c) 196 (d) 16
  37. 37.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
  38. 38.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3575 (c) 4225 (d) 3025
  39. 39.If a + b = 7 and ab = 6, what is (a − b)²? (a) 37 (b) 49 (c) 25 (d) 5
  40. 40.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 1 and 48 (c) 2 and 24 (d) 1 and 13

Answer key

  1. 52
  2. (x + 8)(x + 9)
  3. 41
  4. 1444
  5. 157
  6. (x + 5)(x + 6)
  7. 45
  8. 1
  9. 2809
  10. 48
  11. 17
  12. 146
  13. 9604
  14. 5 and 6
  15. 10
  16. 6084
  17. 6
  18. 2 and 5
  19. (x + 2)(x + 6)
  20. 1849
  21. (x + 2)(x + 3)
  22. 202
  23. 72
  24. 4
  25. (x + 4)(x + 7)
  26. (x + 7)(x + 8)
  27. 2 and 8
  28. 104
  29. 4 and 7
  30. 25
  31. 100
  32. 3 and 8
  33. (x + 4)(x + 5)
  34. (x + 4)(x + 9)
  35. 25
  36. 16
  37. 441
  38. 3025
  39. 25
  40. 6 and 8

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