Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 775 · 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 34²? (a) 1156 (b) 916 (c) 1166 (d) 676
2. 2.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 451 (c) 361 (d) 441
3. 3.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3979 (c) 3969 (d) 3249
4. 4.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
5. 5.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4875 (c) 5625 (d) 4225
6. 6.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 196 (c) 16 (d) 4
7. 7.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2601 (c) 2611 (d) 2401
8. 8.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 421 (c) 157 (d) 289
9. 9.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 1 and 8 (c) 2 and 9 (d) 3 and 6
10. 10.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 841 (c) 851 (d) 899
11. 11.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3731 (c) 3721 (d) 3481
12. 12.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 1 (c) 85 (d) 169
13. 13.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
14. 14.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1531 (c) 1599 (d) 1521
15. 15.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
16. 16.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 1 and 6 (d) 2 and 6
17. 17.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6404 (c) 6724 (d) 6734
18. 18.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 1156 (c) 676 (d) 686
19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
20. 20.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 20 (c) 21 (d) 19
21. 21.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
22. 22.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 1)(x + 10)
23. 23.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2475 (c) 2025 (d) 3025
24. 24.What is the product of the roots of x² − 13x + 40 = 0? (a) 13 (b) 39 (c) 41 (d) 40
25. 25.If a + b = 12 and ab = 11, what is (a − b)²? (a) 122 (b) 144 (c) 100 (d) 10
26. 26.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 289 (d) 539
27. 27.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
28. 28.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x − 3)(x − 5)
29. 29.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 11035 (c) 10025 (d) 9025
30. 30.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 6 (c) 7 (d) 8
31. 31.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
32. 32.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 289 (c) 7 (d) 49
33. 33.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 1 and 11 (c) 1 and 35 (d) 5 and 7
34. 34.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 17 (d) 73
35. 35.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 8659 (c) 7569 (d) 8649
36. 36.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 8075 (c) 7235 (d) 9025
37. 37.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 55 (d) 15
38. 38.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 65 (c) 121 (d) 9
39. 39.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 121 (c) 177 (d) 65
40. 40.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 14 (d) 12

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