Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 134 · 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.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 1 and 32 (c) 2 and 16 (d) 4 and 8
2. 2.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 366 (c) 256 (d) 146
3. 3.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8474 (c) 8104 (d) 8464
4. 4.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)
5. 5.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 56 (d) 16
6. 6.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7406 (c) 7396 (d) 8836
7. 7.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 18 (d) 16
8. 8.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)
9. 9.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
10. 10.If a + b = 10 and ab = 9, what is a² + b²? (a) 91 (b) 118 (c) 82 (d) 100
11. 11.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
12. 12.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 5929 (c) 6391 (d) 6889
13. 13.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 729 (c) 1099 (d) 1089
14. 14.What are the roots of x² − 9x + 20 = 0? (a) 1 and 8 (b) 1 and 20 (c) 4 and 5 (d) 2 and 10
15. 15.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 64 (c) 63 (d) 16
16. 16.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 4 (c) 170 (d) 324
17. 17.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 299 (c) 529 (d) 289
18. 18.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x − 2)(x − 9)
19. 19.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 5 and 6 (c) 1 and 10 (d) 1 and 30
20. 20.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 1 and 15 (c) 3 and 5 (d) 1 and 7
21. 21.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 6) (c) (x − 2)(x − 3) (d) (x + 1)(x + 4)
22. 22.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 80 (c) 144 (d) 208
23. 23.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 72 (d) 73
24. 24.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 4 (c) 5 (d) 7
25. 25.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
26. 26.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 26 (c) 12 (d) 27
27. 27.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 11)
28. 28.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 451 (c) 401 (d) 361
29. 29.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
30. 30.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 1 (c) 271 (d) 181
31. 31.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
32. 32.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 8 (c) 10 (d) 9
33. 33.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 333 (c) 117 (d) 225
34. 34.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 4 (c) 136 (d) 256
35. 35.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
36. 36.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 11 (c) 10 (d) 24
37. 37.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
38. 38.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8464 (c) 7744 (d) 8096
39. 39.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
40. 40.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2116 (c) 2126 (d) 2484

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