Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 886 · 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 = 9 and ab = 20, what is a² + b²? (a) 61 (b) 41 (c) 121 (d) 81
2. 2.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8084 (c) 7406 (d) 8836
3. 3.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 2 and 5 (c) 1 and 8 (d) 1 and 5
4. 4.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
5. 5.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 7) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
6. 6.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5929 (c) 6889 (d) 5939
7. 7.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
8. 8.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 5635 (c) 4225 (d) 4925
9. 9.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
10. 10.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 8659 (c) 8649 (d) 7569
11. 11.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
12. 12.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 144 (c) 4 (d) 2
13. 13.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 133 (c) 241 (d) 97
14. 14.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 10025 (c) 11025 (d) 11035
15. 15.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3364 (c) 3854 (d) 3604
16. 16.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 25 (d) 49
17. 17.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 2 and 20 (c) 5 and 8 (d) 1 and 12
18. 18.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 12 (d) 13
19. 19.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3259 (c) 3249 (d) 3591
20. 20.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
21. 21.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10414 (d) 10404
22. 22.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
23. 23.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2025 (b) 3025 (c) 2475 (d) 2035
24. 24.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 49 (c) 94 (d) 64
25. 25.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 156 (c) 276 (d) 116
26. 26.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1591 (c) 1849 (d) 1379
27. 27.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 63 (d) 64
28. 28.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 8 (c) 18 (d) 9
29. 29.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 225 (c) 657 (d) 441
30. 30.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
31. 31.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5486 (d) 5476
32. 32.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
33. 33.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 4 (c) 2 (d) 64
34. 34.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9226 (c) 9216 (d) 9984
35. 35.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 144 (c) 100 (d) 122
36. 36.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
37. 37.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 100 (c) 4 (d) 2
38. 38.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 1 and 7 (c) 1 and 12 (d) 2 and 6
39. 39.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1454 (c) 1596 (d) 1444
40. 40.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)

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