Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 903 · 40 questions

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.

1. 1.If a + b = 12 and ab = 11, what is a² + b²? (a) 144 (b) 166 (c) 133 (d) 122
2. 2.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 149 (c) 429 (d) 219
3. 3.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
4. 4.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1521 (d) 1601
5. 5.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 494 (c) 324 (d) 484
6. 6.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 8 (c) 9 (d) 10
7. 7.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4225 (c) 3025 (d) 4235
8. 8.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 156 (c) 196 (d) 276
9. 9.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3481 (c) 3731 (d) 3601
10. 10.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 529 (c) 299 (d) 391
11. 11.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 304 (c) 592 (d) 208
12. 12.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 241 (c) 169 (d) 97
13. 13.If a + b = 11 and ab = 10, what is (a − b)²? (a) 9 (b) 101 (c) 81 (d) 121
14. 14.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
15. 15.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 3 and 8 (c) 1 and 24 (d) 1 and 10
16. 16.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 35 (c) 12 (d) 36
17. 17.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
18. 18.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1089 (c) 1099 (d) 729
19. 19.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 899 (c) 961 (d) 851
20. 20.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 13 (c) 9 (d) 5
21. 21.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 12 (c) 13 (d) 11
22. 22.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 185 (d) 273
23. 23.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
24. 24.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3259 (c) 3249 (d) 3591
25. 25.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x − 3)(x − 4)
26. 26.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 10001 (c) 10211 (d) 9801
27. 27.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 17 (c) 9 (d) 25
28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
29. 29.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8464 (c) 8104 (d) 7744
30. 30.What is the product of the roots of x² − 12x + 27 = 0? (a) 26 (b) 12 (c) 27 (d) 28
31. 31.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 164 (c) 324 (d) 4
32. 32.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 3035 (c) 3025 (d) 4225
33. 33.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 5329 (c) 4891 (d) 4489
34. 34.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 10201 (c) 9999 (d) 9801
35. 35.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8291 (d) 8281
36. 36.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
37. 37.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 676 (c) 884 (d) 1156
38. 38.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 117 (c) 3 (d) 9
39. 39.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 9025 (c) 10025 (d) 11025
40. 40.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 4 and 8 (c) 1 and 32 (d) 1 and 11

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