Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 776 · 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 − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 5 (c) 1 and 8 (d) 2 and 4
2. 2.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6391 (c) 5929 (d) 6889
3. 3.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4891 (c) 4499 (d) 4489
4. 4.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 13 (c) 35 (d) 12
5. 5.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6241 (c) 6561 (d) 6399
6. 6.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
7. 7.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3249 (c) 3969 (d) 3979
8. 8.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
9. 9.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 56 (c) 15 (d) 55
10. 10.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 2 and 24 (c) 1 and 13 (d) 6 and 8
11. 11.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 5625 (c) 4235 (d) 4225
12. 12.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1296 (d) 1306
13. 13.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 145 (c) 433 (d) 217
14. 14.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 90 (c) 36 (d) 144
15. 15.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
16. 16.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x + 2)(x + 3) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
17. 17.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
18. 18.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 391 (d) 299
19. 19.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2116 (c) 2516 (d) 2926
20. 20.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 241 (c) 133 (d) 169
21. 21.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 17 (c) 3 (d) 9
22. 22.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1034 (c) 1024 (d) 784
23. 23.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
24. 24.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 144 (c) 104 (d) 184
25. 25.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 3625 (d) 3025
26. 26.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
27. 27.If a + b = 4 and ab = 3, what is a² + b²? (a) 10 (b) 13 (c) 22 (d) 16
28. 28.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 5929 (c) 6409 (d) 6899
29. 29.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
30. 30.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 6 (c) 7 (d) 8
31. 31.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 9801 (c) 10201 (d) 10001
32. 32.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 3)(x + 7)
33. 33.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
34. 34.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8281 (d) 8291
35. 35.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
36. 36.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
37. 37.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 12 (c) 10 (d) 11
38. 38.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10619 (c) 9409 (d) 10609
39. 39.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x − 2)(x − 8) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
40. 40.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 4 (c) 256 (d) 2

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