Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 742 · 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 = 20, what is a² + b²? (a) 81 (b) 61 (c) 41 (d) 121
2. 2.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 10816 (c) 9984 (d) 9216
3. 3.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 384 (c) 266 (d) 256
4. 4.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 85 (d) 127
5. 5.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 9 (c) 3 (d) 185
6. 6.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 961 (c) 851 (d) 841
7. 7.What are the roots of x² − 11x + 18 = 0? (a) 1 and 10 (b) 3 and 6 (c) 1 and 18 (d) 2 and 9
8. 8.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
9. 9.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
10. 10.If a + b = 13 and ab = 36, what is a² + b²? (a) 97 (b) 133 (c) 241 (d) 169
11. 11.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 256 (c) 130 (d) 193
12. 12.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 4 (d) 400
13. 13.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 8 (c) 256 (d) 64
14. 14.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10609 (c) 10009 (d) 10619
15. 15.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1591 (c) 1849 (d) 1369
16. 16.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x + 2)(x + 9) (d) (x − 2)(x − 9)
17. 17.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 289 (c) 157 (d) 421
18. 18.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6241 (c) 6561 (d) 6401
19. 19.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 33 (c) 17 (d) 25
20. 20.If a + b = 8 and ab = 7, what is a² + b²? (a) 57 (b) 78 (c) 50 (d) 64
21. 21.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 971 (c) 961 (d) 901
22. 22.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 8 and 9 (d) 2 and 36
23. 23.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)
24. 24.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
25. 25.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 8659 (c) 7569 (d) 8649
26. 26.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4225 (c) 3025 (d) 4235
27. 27.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 14 (c) 16 (d) 15
28. 28.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 84 (c) 68 (d) 132
29. 29.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)
30. 30.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2116 (c) 2484 (d) 2126
31. 31.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1454 (d) 1444
32. 32.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 9604 (c) 10004 (d) 10414
33. 33.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10001 (c) 10211 (d) 10201
34. 34.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 1 and 13 (c) 3 and 15 (d) 5 and 9
35. 35.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
36. 36.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 1 (c) 113 (d) 169
37. 37.What is the product of the roots of x² − 13x + 36 = 0? (a) 36 (b) 13 (c) 37 (d) 35
38. 38.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 23 (c) 25 (d) 11
39. 39.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 7 (c) 4 (d) 5
40. 40.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1296 (c) 1936 (d) 1584

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