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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 982 · 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 product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 22 (c) 10 (d) 21
  2. 2.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2809 (c) 2509 (d) 2819
  3. 3.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 325 (c) 225 (d) 175
  4. 4.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
  5. 5.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 4096 (c) 3584 (d) 3136
  6. 6.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 57 (c) 15 (d) 55
  7. 7.If a + b = 13 and ab = 12, what is (a − b)²? (a) 145 (b) 11 (c) 169 (d) 121
  8. 8.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)
  9. 9.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
  10. 10.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
  11. 11.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 43 (c) 13 (d) 41
  12. 12.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 1)(x + 11)
  13. 13.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 3 (c) 9 (d) 361
  14. 14.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x + 2)(x + 8)
  15. 15.If a + b = 14 and ab = 48, what is a² + b²? (a) 292 (b) 148 (c) 100 (d) 196
  16. 16.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 635 (c) 625 (d) 875
  17. 17.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x + 1)(x + 6) (d) (x − 2)(x − 3)
  18. 18.If a + b = 17 and ab = 66, what is (a − b)²? (a) 25 (b) 289 (c) 157 (d) 5
  19. 19.What are the roots of x² − 10x + 24 = 0? (a) 2 and 12 (b) 4 and 6 (c) 1 and 9 (d) 1 and 24
  20. 20.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 16 (c) 196 (d) 4
  21. 21.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
  22. 22.If a + b = 14 and ab = 33, what is a² + b²? (a) 163 (b) 262 (c) 196 (d) 130
  23. 23.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 11 (c) 10 (d) 21
  24. 24.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 421 (d) 223
  25. 25.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1936 (c) 1946 (d) 1616
  26. 26.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
  27. 27.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1166 (d) 1156
  28. 28.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 1)(x + 24)
  29. 29.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 289 (c) 3 (d) 149
  30. 30.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 11 (c) 13 (d) 12
  31. 31.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)
  32. 32.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 133 (c) 166 (d) 122
  33. 33.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 214 (c) 144 (d) 74
  34. 34.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 52 (c) 100 (d) 4
  35. 35.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 2025 (d) 3035
  36. 36.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
  37. 37.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 14 (c) 8 (d) 16
  38. 38.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
  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.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 2 and 6 (c) 1 and 7 (d) 1 and 15

Answer key

  1. 21
  2. 2809
  3. 125
  4. 17
  5. 3136
  6. 56
  7. 121
  8. (x + 5)(x + 6)
  9. (x + 8)(x + 9)
  10. (x + 4)(x + 5)
  11. 42
  12. (x + 3)(x + 9)
  13. 9
  14. (x + 2)(x + 8)
  15. 100
  16. 625
  17. (x + 2)(x + 3)
  18. 25
  19. 4 and 6
  20. 16
  21. (x + 6)(x + 8)
  22. 130
  23. 10
  24. 157
  25. 1936
  26. (x + 6)(x + 7)
  27. 1156
  28. (x + 4)(x + 6)
  29. 9
  30. 12
  31. (x + 5)(x + 8)
  32. 122
  33. 74
  34. 4
  35. 3025
  36. (x + 7)(x + 8)
  37. 15
  38. 72
  39. (x + 2)(x + 9)
  40. 3 and 5

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