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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 425 · 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.What is the sum of the roots of x² − 9x + 14 = 0? (a) 9 (b) 8 (c) 10 (d) 14
  2. 2.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 10 (d) 11
  3. 3.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 12 (c) 7 (d) 13
  4. 4.If a + b = 11 and ab = 18, what is a² + b²? (a) 103 (b) 85 (c) 121 (d) 157
  5. 5.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
  6. 6.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 10 (c) 12 (d) 11
  7. 7.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
  8. 8.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 249 (c) 129 (d) 169
  9. 9.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
  10. 10.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1089 (c) 729 (d) 1099
  11. 11.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
  12. 12.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 57 (c) 55 (d) 56
  13. 13.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 5776 (c) 7056 (d) 6384
  14. 14.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 271 (d) 541
  15. 15.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 971 (d) 961
  16. 16.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 122 (c) 144 (d) 100
  17. 17.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
  18. 18.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
  19. 19.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4624 (c) 4896 (d) 4634
  20. 20.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 145 (c) 289 (d) 217
  21. 21.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x + 3)(x + 7) (d) (x − 3)(x − 7)
  22. 22.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
  23. 23.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 84 (c) 100 (d) 132
  24. 24.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8099 (c) 8281 (d) 7931
  25. 25.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)
  26. 26.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 1 and 56 (d) 2 and 28
  27. 27.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 14 (c) 13 (d) 15
  28. 28.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 2 and 4 (c) 1 and 5 (d) 1 and 8
  29. 29.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2209 (c) 2491 (d) 2219
  30. 30.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6561 (c) 6571 (d) 6241
  31. 31.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
  32. 32.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 10016 (c) 10826 (d) 9216
  33. 33.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3979 (b) 3249 (c) 3609 (d) 3969
  34. 34.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2916 (c) 2516 (d) 2116
  35. 35.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 46 (d) 44
  36. 36.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 85 (c) 127 (d) 169
  37. 37.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 125 (c) 169 (d) 81
  38. 38.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 416 (c) 586 (d) 256
  39. 39.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
  40. 40.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 13 (c) 14 (d) 42

Answer key

  1. 9
  2. 11
  3. 12
  4. 85
  5. 16
  6. 11
  7. (x + 7)(x + 9)
  8. 89
  9. (x + 5)(x + 6)
  10. 1089
  11. 1
  12. 56
  13. 5776
  14. 181
  15. 961
  16. 100
  17. (x + 6)(x + 7)
  18. (x + 5)(x + 8)
  19. 4624
  20. 145
  21. (x + 3)(x + 7)
  22. (x + 3)(x + 4)
  23. 68
  24. 7921
  25. (x + 8)(x + 9)
  26. 7 and 8
  27. 14
  28. 2 and 4
  29. 2209
  30. 6561
  31. (x + 2)(x + 7)
  32. 10816
  33. 3969
  34. 2916
  35. 45
  36. 85
  37. 81
  38. 576
  39. 8 and 9
  40. 13

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