Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 852 · 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 = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 4 (d) 16
2. 2.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 146 (c) 36 (d) 256
3. 3.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 961 (d) 901
4. 4.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8104 (d) 8464
5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
6. 6.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 100 (c) 91 (d) 118
7. 7.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
8. 8.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1306 (c) 1936 (d) 1584
9. 9.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 541 (d) 271
10. 10.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 10 (c) 11 (d) 24
11. 11.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 7235 (c) 8075 (d) 9025
12. 12.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 425 (c) 225 (d) 625
13. 13.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 52 (c) 88 (d) 40
14. 14.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 52 (c) 20 (d) 36
15. 15.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 39 (c) 41 (d) 40
16. 16.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
17. 17.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
18. 18.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)
19. 19.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9801 (c) 9811 (d) 9999
20. 20.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 576 (d) 416
21. 21.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 2 and 10 (c) 1 and 27 (d) 1 and 11
22. 22.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 5 (c) 169 (d) 25
23. 23.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
24. 24.What are the roots of x² − 7x + 10 = 0? (a) 1 and 6 (b) 2 and 5 (c) 1 and 10 (d) 3 and 4
25. 25.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1681 (c) 1599 (d) 1531
26. 26.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7406 (c) 7396 (d) 8836
27. 27.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 11 (c) 12 (d) 10
28. 28.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2484 (c) 2126 (d) 2116
29. 29.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 851 (c) 841 (d) 961
30. 30.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 1 and 30 (d) 5 and 6
31. 31.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 11 (c) 27 (d) 28
32. 32.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 1)(x + 6) (d) (x + 2)(x + 5)
33. 33.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 794 (c) 1024 (d) 784
34. 34.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 739 (c) 891 (d) 729
35. 35.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 2 and 18 (c) 4 and 9 (d) 1 and 12
36. 36.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
37. 37.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 72 (d) 16
38. 38.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 3 (c) 289 (d) 149
39. 39.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
40. 40.If a + b = 18 and ab = 80, what is (a − b)²? (a) 4 (b) 2 (c) 164 (d) 324

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