Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 769 · 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 − 11x + 18 = 0? (a) 3 and 6 (b) 2 and 9 (c) 1 and 18 (d) 1 and 10
2. 2.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 273 (c) 185 (d) 361
3. 3.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 25 (c) 5 (d) 361
4. 4.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
5. 5.What are the roots of x² − 8x + 15 = 0? (a) 3 and 5 (b) 1 and 15 (c) 1 and 7 (d) 2 and 6
6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
7. 7.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 539 (d) 529
8. 8.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 4096 (c) 3136 (d) 3584
9. 9.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2704 (c) 2714 (d) 2504
10. 10.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 2 (d) 202
11. 11.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 6889 (c) 5939 (d) 6391
12. 12.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 18) (c) (x + 1)(x + 10) (d) (x + 2)(x + 9)
13. 13.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 73 (c) 25 (d) 49
14. 14.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
15. 15.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
16. 16.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 2525 (d) 2025
17. 17.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 7921 (c) 8101 (d) 8291
18. 18.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 49 (c) 65 (d) 7
19. 19.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
20. 20.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 42 (d) 43
21. 21.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 144 (c) 208 (d) 80
22. 22.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 324 (c) 396 (d) 484
23. 23.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9811 (c) 10201 (d) 9999
24. 24.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 149 (c) 219 (d) 289
25. 25.What is the product of the roots of x² − 14x + 45 = 0? (a) 46 (b) 14 (c) 45 (d) 44
26. 26.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 56 (c) 1 and 14 (d) 2 and 28
27. 27.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
28. 28.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 10 (c) 13 (d) 22
29. 29.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 10 (c) 24 (d) 12
30. 30.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 9409 (c) 10009 (d) 10619
31. 31.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 400 (c) 304 (d) 592
32. 32.If a + b = 9 and ab = 18, what is (a − b)²? (a) 81 (b) 3 (c) 45 (d) 9
33. 33.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 961 (d) 971
34. 34.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2491 (b) 2809 (c) 2209 (d) 2219
35. 35.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
36. 36.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)
37. 37.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 16 (c) 63 (d) 17
38. 38.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 26 (c) 12 (d) 27
39. 39.If a + b = 11 and ab = 24, what is a² + b²? (a) 97 (b) 121 (c) 73 (d) 169
40. 40.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 686 (c) 1156 (d) 676

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