Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 20 · 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 = 9 and ab = 14, what is a² + b²? (a) 67 (b) 81 (c) 53 (d) 109
2. 2.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3249 (c) 3969 (d) 3259
3. 3.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4096 (c) 3136 (d) 4106
4. 4.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 169 (c) 25 (d) 97
5. 5.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
6. 6.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 841 (d) 961
7. 7.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
8. 8.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 484 (d) 494
9. 9.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)
10. 10.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 337 (d) 113
11. 11.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 63 (c) 45 (d) 117
12. 12.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 15 (c) 54 (d) 16
13. 13.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 25 (c) 5 (d) 361
14. 14.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 27 (c) 12 (d) 26
15. 15.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 6409 (c) 5929 (d) 6899
16. 16.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x + 1)(x + 10) (c) (x − 3)(x − 8) (d) (x + 1)(x + 24)
17. 17.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1306 (c) 1296 (d) 1936
18. 18.If a + b = 5 and ab = 6, what is a² + b²? (a) 13 (b) 37 (c) 19 (d) 25
19. 19.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2916 (c) 2116 (d) 2126
20. 20.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 2 (c) 324 (d) 4
21. 21.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
22. 22.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x − 4)(x − 7) (d) (x + 1)(x + 28)
23. 23.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 729 (c) 1099 (d) 1089
24. 24.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
25. 25.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 7 (c) 65 (d) 49
26. 26.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 1 and 63 (d) 3 and 21
27. 27.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1849 (c) 1591 (d) 1369
28. 28.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
29. 29.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 52 (c) 20 (d) 28
30. 30.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1609 (d) 1849
31. 31.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 277 (c) 529 (d) 193
32. 32.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 11 (c) 30 (d) 31
33. 33.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2601 (c) 2499 (d) 2411
34. 34.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3844 (c) 3596 (d) 3364
35. 35.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 625 (c) 225 (d) 635
36. 36.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 16 (c) 22 (d) 10
37. 37.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)
38. 38.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
39. 39.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 1 and 12 (c) 5 and 8 (d) 2 and 20
40. 40.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 4899 (c) 4761 (d) 5041

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