Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 833 · 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 = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 117 (c) 3 (d) 9
2. 2.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1235 (c) 1225 (d) 2025
3. 3.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5194 (c) 4624 (d) 5184
4. 4.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 55 (d) 56
5. 5.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)
6. 6.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2304 (c) 2314 (d) 2496
7. 7.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 2025 (d) 3035
8. 8.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1601 (c) 1681 (d) 1691
9. 9.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 9 (c) 8 (d) 14
10. 10.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 25 (c) 121 (d) 5
11. 11.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 11 (c) 27 (d) 12
12. 12.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1379 (c) 1369 (d) 1591
13. 13.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 635 (b) 625 (c) 225 (d) 425
14. 14.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 739 (c) 1089 (d) 891
15. 15.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 1 and 12 (c) 2 and 18 (d) 4 and 9
16. 16.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 9 (c) 16 (d) 10
17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
18. 18.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 46 (c) 26 (d) 31
19. 19.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2025 (b) 1625 (c) 1225 (d) 2035
20. 20.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 63 (d) 16
21. 21.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 10) (c) (x + 1)(x + 28) (d) (x − 4)(x − 7)
22. 22.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 19 (c) 13 (d) 25
23. 23.If a + b = 19 and ab = 84, what is (a − b)²? (a) 193 (b) 5 (c) 25 (d) 361
24. 24.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 7 (c) 8 (d) 12
25. 25.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 10 (c) 18 (d) 12
26. 26.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 14 (c) 13 (d) 40
27. 27.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 121 (c) 9 (d) 3
28. 28.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 72 (d) 18
29. 29.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 11 (c) 12 (d) 10
30. 30.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 49 (c) 85 (d) 121
31. 31.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7406 (c) 7396 (d) 8836
32. 32.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 301 (c) 598 (d) 400
33. 33.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 42 (d) 13
34. 34.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 144 (c) 112 (d) 80
35. 35.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 2 and 6 (c) 1 and 12 (d) 3 and 4
36. 36.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
37. 37.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
38. 38.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 289 (c) 421 (d) 157
39. 39.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 361 (c) 401 (d) 451
40. 40.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 100 (c) 132 (d) 84

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