Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 105 · 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 28²? (a) 1024 (b) 896 (c) 784 (d) 794
2. 2.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 4 (c) 2 (d) 196
3. 3.What are the roots of x² − 7x + 12 = 0? (a) 1 and 12 (b) 1 and 6 (c) 3 and 4 (d) 2 and 6
4. 4.If a + b = 11 and ab = 24, what is (a − b)²? (a) 5 (b) 121 (c) 25 (d) 73
5. 5.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1296 (c) 1936 (d) 1584
6. 6.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
7. 7.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 25 (c) 19 (d) 13
8. 8.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2819 (c) 2509 (d) 2809
9. 9.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 361 (c) 25 (d) 5
10. 10.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8109 (c) 7569 (d) 8649
11. 11.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3591 (b) 3969 (c) 3249 (d) 3259
12. 12.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 62 (d) 63
13. 13.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 676 (c) 686 (d) 1156
14. 14.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 10004 (c) 10404 (d) 9604
15. 15.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10619 (d) 10609
16. 16.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
17. 17.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 1 and 14 (c) 6 and 9 (d) 2 and 27
18. 18.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 256 (d) 136
19. 19.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 4)(x + 7) (c) (x − 4)(x − 7) (d) (x + 1)(x + 10)
20. 20.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x − 4)(x − 8)
21. 21.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 53 (d) 54
22. 22.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 3)(x + 4) (d) (x + 1)(x + 6)
23. 23.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 14 (c) 9 (d) 8
24. 24.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 81 (c) 9 (d) 45
25. 25.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 10 (c) 16 (d) 13
26. 26.If a + b = 13 and ab = 12, what is (a − b)²? (a) 11 (b) 169 (c) 121 (d) 145
27. 27.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
28. 28.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
29. 29.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 64 (c) 88 (d) 52
30. 30.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
31. 31.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8281 (c) 7921 (d) 8101
32. 32.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
33. 33.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 1 and 13 (c) 6 and 8 (d) 2 and 24
34. 34.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 49 (c) 73 (d) 37
35. 35.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)
36. 36.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
37. 37.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 17 (c) 18 (d) 9
38. 38.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 256 (d) 130
39. 39.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2116 (c) 2126 (d) 2916
40. 40.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 225 (c) 137 (d) 313

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