Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 570 · 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 = 6 and ab = 8, what is a² + b²? (a) 52 (b) 36 (c) 28 (d) 20
2. 2.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 625 (c) 225 (d) 375
3. 3.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x − 4)(x − 6) (d) (x + 4)(x + 6)
4. 4.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
5. 5.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1454 (c) 1764 (d) 1444
6. 6.If a + b = 10 and ab = 24, what is a² + b²? (a) 100 (b) 76 (c) 148 (d) 52
7. 7.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 121 (c) 61 (d) 1
8. 8.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 100 (c) 91 (d) 118
9. 9.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 729 (d) 1089
10. 10.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 12 (d) 14
11. 11.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
12. 12.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4925 (b) 5635 (c) 4225 (d) 5625
13. 13.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 14 (c) 6 and 9 (d) 1 and 54
14. 14.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)
15. 15.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 16 (c) 170 (d) 4
16. 16.What is the product of the roots of x² − 13x + 36 = 0? (a) 36 (b) 37 (c) 35 (d) 13
17. 17.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6724 (c) 6094 (d) 6084
18. 18.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
19. 19.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6899 (c) 6889 (d) 6409
20. 20.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
21. 21.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1369 (c) 1379 (d) 1849
22. 22.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)
23. 23.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2126 (c) 2916 (d) 2484
24. 24.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 529 (c) 361 (d) 193
25. 25.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 9 (c) 11 (d) 10
26. 26.If a + b = 12 and ab = 35, what is (a − b)²? (a) 74 (b) 2 (c) 144 (d) 4
27. 27.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
28. 28.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 7056 (c) 7066 (d) 6416
29. 29.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 841 (c) 971 (d) 901
30. 30.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
31. 31.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 54 (c) 15 (d) 53
32. 32.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1609 (c) 1369 (d) 1849
33. 33.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 244 (c) 148 (d) 196
34. 34.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
35. 35.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 5 and 8 (c) 2 and 20 (d) 1 and 40
36. 36.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 1 and 30 (d) 5 and 6
37. 37.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 13 (c) 14 (d) 15
38. 38.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
39. 39.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 94 (c) 49 (d) 34
40. 40.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)

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