Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 100 · 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.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1369 (d) 1849
2. 2.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4891 (c) 4499 (d) 4489
3. 3.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 13 (d) 14
4. 4.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 25 (c) 1 (d) 13
5. 5.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6561 (c) 6401 (d) 6571
6. 6.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2116 (c) 2516 (d) 2916
7. 7.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 1034 (d) 784
8. 8.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
9. 9.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 676 (c) 1156 (d) 1166
10. 10.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 1 (c) 61 (d) 41
11. 11.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 8075 (c) 7225 (d) 9025
12. 12.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 5625 (d) 7225
13. 13.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
14. 14.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8101 (c) 8291 (d) 8281
15. 15.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 1)(x + 10) (d) (x + 2)(x + 9)
16. 16.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
17. 17.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 127 (c) 253 (d) 85
18. 18.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 3 (c) 9 (d) 441
19. 19.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 11 (c) 10 (d) 9
20. 20.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 7 (c) 9 (d) 10
21. 21.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
22. 22.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
23. 23.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 10816 (c) 9226 (d) 9216
24. 24.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
25. 25.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 15 (c) 54 (d) 55
26. 26.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 97 (c) 169 (d) 73
27. 27.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
28. 28.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 729 (d) 1089
29. 29.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 2 and 24 (c) 1 and 13 (d) 1 and 48
30. 30.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 13 (c) 25 (d) 19
31. 31.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 971 (c) 901 (d) 961
32. 32.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 15 (c) 14 (d) 8
33. 33.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 371 (c) 399 (d) 361
34. 34.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 1)(x + 6)
35. 35.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 45 (c) 81 (d) 9
36. 36.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 7396 (c) 8836 (d) 8084
37. 37.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 68 (c) 100 (d) 132
38. 38.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
39. 39.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 34 (c) 4 (d) 2
40. 40.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 1)(x + 11)

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