Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 945 · 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 82²? (a) 6084 (b) 6404 (c) 6734 (d) 6724
2. 2.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5194 (c) 4904 (d) 5184
3. 3.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 333 (c) 117 (d) 171
4. 4.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 4 (c) 256 (d) 130
5. 5.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
6. 6.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
7. 7.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 72 (c) 18 (d) 17
8. 8.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 1089 (d) 729
9. 9.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 169 (c) 127 (d) 1
10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
11. 11.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
12. 12.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 219 (c) 289 (d) 429
13. 13.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 6889 (c) 5929 (d) 5939
14. 14.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
15. 15.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271
16. 16.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 2 (d) 202
17. 17.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 8464 (c) 7744 (d) 8104
18. 18.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 35 (c) 34 (d) 12
19. 19.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 1 and 7 (c) 2 and 6 (d) 1 and 15
20. 20.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
21. 21.What are the roots of x² − 12x + 35 = 0? (a) 5 and 7 (b) 1 and 11 (c) 2 and 10 (d) 1 and 35
22. 22.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 14 (c) 42 (d) 12
23. 23.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 6 (c) 7 (d) 12
24. 24.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1681 (c) 1531 (d) 1599
25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
26. 26.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3854 (c) 3364 (d) 3604
27. 27.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 244 (c) 172 (d) 148
28. 28.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 43 (d) 42
29. 29.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3146 (c) 4096 (d) 3584
30. 30.What is the product of the roots of x² − 7x + 12 = 0? (a) 13 (b) 11 (c) 12 (d) 7
31. 31.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
32. 32.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2116 (c) 2484 (d) 2126
33. 33.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2809 (b) 2209 (c) 2509 (d) 2819
34. 34.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 12 (c) 28 (d) 26
35. 35.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 16 (d) 63
36. 36.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
37. 37.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
38. 38.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 537 (c) 361 (d) 273
39. 39.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1444 (c) 1774 (d) 1604
40. 40.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1575 (d) 1235

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