Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 13 · 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 72²? (a) 5194 (b) 5184 (c) 4904 (d) 4624
2. 2.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 484 (c) 494 (d) 404
3. 3.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)
4. 4.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
5. 5.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 361 (c) 399 (d) 371
6. 6.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
7. 7.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 4 (c) 36 (d) 20
8. 8.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 5 (c) 225 (d) 125
9. 9.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 3 and 6 (c) 1 and 14 (d) 2 and 7
10. 10.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)
11. 11.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 225 (c) 441 (d) 3
12. 12.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2126 (c) 2916 (d) 2116
13. 13.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)
14. 14.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 5929 (c) 6899 (d) 6409
15. 15.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)
16. 16.If a + b = 12 and ab = 27, what is (a − b)²? (a) 6 (b) 36 (c) 144 (d) 90
17. 17.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 7 (c) 49 (d) 289
18. 18.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 9 (c) 45 (d) 3
19. 19.If a + b = 14 and ab = 48, what is (a − b)²? (a) 2 (b) 196 (c) 4 (d) 100
20. 20.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
21. 21.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 7 (d) 5
22. 22.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
23. 23.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 117 (c) 9 (d) 225
24. 24.What are the roots of x² − 12x + 32 = 0? (a) 1 and 32 (b) 1 and 11 (c) 4 and 8 (d) 2 and 16
25. 25.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 409 (c) 539 (d) 529
26. 26.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 58 (d) 4
27. 27.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 225 (c) 175 (d) 325
28. 28.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 65 (d) 177
29. 29.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x − 5)(x − 7) (d) (x + 1)(x + 11)
30. 30.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)
31. 31.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 361 (d) 193
32. 32.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 7 (c) 1 and 15 (d) 3 and 5
33. 33.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 400 (d) 598
34. 34.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 24 (c) 25 (d) 11
35. 35.If a + b = 13 and ab = 36, what is a² + b²? (a) 241 (b) 133 (c) 97 (d) 169
36. 36.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 925 (c) 625 (d) 1225
37. 37.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1575 (c) 1225 (d) 2025
38. 38.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 576 (c) 266 (d) 256
39. 39.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1369 (c) 1609 (d) 1849
40. 40.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6889 (c) 6391 (d) 5929

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