Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 397 · 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 = 8 and ab = 12, what is (a − b)²? (a) 40 (b) 4 (c) 16 (d) 64
2. 2.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
3. 3.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
4. 4.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 10 (c) 16 (d) 11
5. 5.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 361 (c) 5 (d) 193
6. 6.If a + b = 11 and ab = 24, what is (a − b)²? (a) 121 (b) 5 (c) 25 (d) 73
7. 7.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 196 (c) 256 (d) 376
8. 8.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 625 (c) 1225 (d) 875
9. 9.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
10. 10.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3844 (c) 3854 (d) 3604
11. 11.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 1 and 6 (c) 2 and 4 (d) 2 and 3
12. 12.If a + b = 5 and ab = 4, what is (a − b)²? (a) 3 (b) 9 (c) 25 (d) 17
13. 13.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 13 (c) 43 (d) 42
14. 14.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 1)(x + 11)
15. 15.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 484 (c) 324 (d) 334
16. 16.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5476 (d) 5486
17. 17.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 441 (c) 451 (d) 361
18. 18.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
19. 19.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
20. 20.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 256 (c) 201 (d) 366
21. 21.What is the sum of the roots of x² − 15x + 54 = 0? (a) 15 (b) 16 (c) 54 (d) 14
22. 22.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8281 (c) 8101 (d) 8291
23. 23.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 15 (c) 14 (d) 45
24. 24.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6561 (c) 6251 (d) 6399
25. 25.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 225 (c) 3 (d) 117
26. 26.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3979 (c) 3249 (d) 3969
27. 27.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
28. 28.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
29. 29.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 10 (c) 20 (d) 9
30. 30.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9996 (b) 9604 (c) 9614 (d) 10404
31. 31.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 46 (c) 36 (d) 26
32. 32.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 142 (c) 58 (d) 100
33. 33.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4909 (c) 5329 (d) 4489
34. 34.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 49 (c) 7 (d) 289
35. 35.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 2 (c) 4 (d) 130
36. 36.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 10) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
37. 37.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3625 (c) 3025 (d) 4225
38. 38.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 196 (c) 172 (d) 148
39. 39.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 1)(x + 7)
40. 40.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 7225 (d) 5625

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