Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 954 · 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.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 10 (c) 2 and 5 (d) 1 and 6
2. 2.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 8075 (b) 9025 (c) 7225 (d) 7235
3. 3.If a + b = 8 and ab = 12, what is a² + b²? (a) 64 (b) 52 (c) 40 (d) 88
4. 4.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 2)(x + 9) (c) (x + 1)(x + 10) (d) (x + 1)(x + 18)
5. 5.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 73 (c) 71 (d) 72
6. 6.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 2025 (c) 1575 (d) 1225
7. 7.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 16 (c) 11 (d) 9
8. 8.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 81 (c) 45 (d) 9
9. 9.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 10001 (d) 9801
10. 10.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 961 (c) 899 (d) 851
11. 11.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
12. 12.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)
13. 13.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 1225 (c) 635 (d) 625
14. 14.If a + b = 7 and ab = 6, what is (a − b)²? (a) 49 (b) 37 (c) 5 (d) 25
15. 15.If a + b = 10 and ab = 16, what is a² + b²? (a) 132 (b) 84 (c) 68 (d) 100
16. 16.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 4106 (c) 3136 (d) 3616
17. 17.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 375 (c) 225 (d) 235
18. 18.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
19. 19.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1531 (b) 1521 (c) 1599 (d) 1681
20. 20.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 181 (d) 271
21. 21.If a + b = 13 and ab = 22, what is a² + b²? (a) 125 (b) 169 (c) 213 (d) 147
22. 22.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 657 (c) 225 (d) 441
23. 23.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7056 (c) 5776 (d) 7066
24. 24.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 109 (c) 139 (d) 229
25. 25.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
26. 26.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2025 (c) 2525 (d) 3025
27. 27.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)
28. 28.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 6 and 9 (c) 2 and 27 (d) 1 and 14
29. 29.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 33 (c) 17 (d) 25
30. 30.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 54 (d) 14
31. 31.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)
32. 32.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 7 and 9 (d) 1 and 63
33. 33.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2611 (b) 2401 (c) 2601 (d) 2501
34. 34.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)
35. 35.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 169 (c) 121 (d) 145
36. 36.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 7 and 8 (d) 1 and 56
37. 37.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 36 (c) 46 (d) 26
38. 38.If a + b = 10 and ab = 21, what is (a − b)²? (a) 58 (b) 100 (c) 4 (d) 16
39. 39.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2714 (d) 2504
40. 40.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 1 and 14 (c) 1 and 8 (d) 3 and 6

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