Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 629 · 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 87²? (a) 7569 (b) 8091 (c) 8649 (d) 7579
2. 2.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
3. 3.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10619 (c) 10009 (d) 10609
4. 4.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2809 (b) 2491 (c) 2219 (d) 2209
5. 5.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202
6. 6.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 8 (c) 7 (d) 10
7. 7.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1296 (c) 1936 (d) 1616
8. 8.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
9. 9.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 49 (c) 37 (d) 1
10. 10.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 14 (c) 8 (d) 9
11. 11.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 14 (c) 2 and 27 (d) 1 and 54
12. 12.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 84 (c) 68 (d) 132
13. 13.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 52 (c) 100 (d) 2
14. 14.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 5041 (c) 4761 (d) 4901
15. 15.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 8125 (d) 7225
16. 16.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 1089 (c) 891 (d) 729
17. 17.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)
18. 18.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6084 (c) 6724 (d) 6404
19. 19.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 100 (c) 292 (d) 148
20. 20.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 37 (c) 19 (d) 13
21. 21.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 7569 (c) 8659 (d) 8109
22. 22.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
23. 23.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 25 (c) 17 (d) 33
24. 24.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 10 (c) 11 (d) 24
25. 25.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
26. 26.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 56 (d) 55
27. 27.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
28. 28.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 16 (c) 14 (d) 54
29. 29.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 529 (c) 409 (d) 289
30. 30.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 25 (c) 5 (d) 97
31. 31.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 15 (c) 14 (d) 13
32. 32.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 13 (c) 12 (d) 40
33. 33.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
34. 34.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 4096 (c) 3136 (d) 3584
35. 35.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 909 (c) 1099 (d) 1089
36. 36.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 16 (d) 1 and 72
37. 37.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 8 (c) 9 (d) 10
38. 38.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 7 (d) 8
39. 39.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
40. 40.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7225 (b) 5625 (c) 7235 (d) 6425

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