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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 423 · 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 = 13 and ab = 42, what is a² + b²? (a) 127 (b) 85 (c) 253 (d) 169
  2. 2.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
  3. 3.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3481 (c) 3599 (d) 3721
  4. 4.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 13 (c) 7 (d) 9
  5. 5.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2809 (c) 2209 (d) 2509
  6. 6.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
  7. 7.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
  8. 8.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)
  9. 9.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 9 (c) 25 (d) 17
  10. 10.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2035 (c) 2025 (d) 3025
  11. 11.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 16 (d) 15
  12. 12.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 169 (d) 225
  13. 13.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 163 (c) 196 (d) 130
  14. 14.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 144 (c) 112 (d) 80
  15. 15.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 598 (d) 400
  16. 16.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 225 (c) 337 (d) 113
  17. 17.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 13 (c) 12 (d) 35
  18. 18.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
  19. 19.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
  20. 20.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 81 (c) 9 (d) 169
  21. 21.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 884 (d) 686
  22. 22.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 7754 (c) 8096 (d) 8464
  23. 23.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 256 (c) 196 (d) 136
  24. 24.If a + b = 11 and ab = 10, what is (a − b)²? (a) 101 (b) 121 (c) 9 (d) 81
  25. 25.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 55 (d) 57
  26. 26.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11025 (c) 11035 (d) 10025
  27. 27.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
  28. 28.What is the product of the roots of x² − 10x + 21 = 0? (a) 21 (b) 10 (c) 22 (d) 20
  29. 29.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 19 (c) 17 (d) 11
  30. 30.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 41 (c) 40 (d) 39
  31. 31.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)
  32. 32.If a + b = 17 and ab = 66, what is (a − b)²? (a) 157 (b) 25 (c) 289 (d) 5
  33. 33.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 10404 (c) 9604 (d) 9614
  34. 34.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 625 (c) 1225 (d) 635
  35. 35.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 304 (c) 592 (d) 208
  36. 36.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 1 (c) 127 (d) 85
  37. 37.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 36 (c) 52 (d) 20
  38. 38.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 45 (c) 46 (d) 14
  39. 39.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 100 (c) 4 (d) 2
  40. 40.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 8 (d) 10

Answer key

  1. 85
  2. (x + 6)(x + 7)
  3. 3481
  4. 5
  5. 2809
  6. (x + 5)(x + 7)
  7. (x + 3)(x + 8)
  8. (x + 6)(x + 8)
  9. 9
  10. 2025
  11. 16
  12. 1
  13. 130
  14. 80
  15. 202
  16. 113
  17. 12
  18. (x + 4)(x + 9)
  19. (x + 7)(x + 8)
  20. 81
  21. 676
  22. 7744
  23. 136
  24. 81
  25. 56
  26. 11025
  27. (x + 3)(x + 4)
  28. 21
  29. 18
  30. 40
  31. (x + 2)(x + 7)
  32. 25
  33. 9604
  34. 625
  35. 208
  36. 1
  37. 20
  38. 45
  39. 4
  40. 9

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