Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 659 · 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 64²? (a) 3616 (b) 3136 (c) 4096 (d) 4106
2. 2.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 25 (c) 19 (d) 13
3. 3.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 16 (c) 4 (d) 10
4. 4.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 2 and 10 (c) 1 and 11 (d) 3 and 9
5. 5.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 225 (d) 337
6. 6.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 19 (c) 18 (d) 17
7. 7.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 109 (c) 74 (d) 144
8. 8.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4366 (c) 4356 (d) 4884
9. 9.If a + b = 11 and ab = 24, what is (a − b)²? (a) 5 (b) 121 (c) 25 (d) 73
10. 10.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 11 (c) 30 (d) 12
11. 11.If a + b = 12 and ab = 20, what is (a − b)²? (a) 104 (b) 8 (c) 64 (d) 144
12. 12.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
13. 13.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 16 (c) 15 (d) 56
14. 14.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 884 (c) 676 (d) 686
15. 15.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 169 (c) 241 (d) 97
16. 16.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1616 (c) 1946 (d) 1936
17. 17.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 7 (c) 6 (d) 5
18. 18.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 6)(x + 8)
19. 19.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 8 (c) 5 (d) 7
20. 20.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 598 (d) 202
21. 21.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
22. 22.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 41 (c) 61 (d) 121
23. 23.If a + b = 10 and ab = 9, what is a² + b²? (a) 91 (b) 82 (c) 118 (d) 100
24. 24.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x − 3)(x − 8) (d) (x + 3)(x + 8)
25. 25.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x − 2)(x − 7)
26. 26.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5476 (d) 5486
27. 27.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x − 3)(x − 5)
28. 28.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2926 (c) 2916 (d) 2116
29. 29.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 14 (c) 47 (d) 48
30. 30.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10619 (c) 10609 (d) 9409
31. 31.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 21 (c) 3 and 7 (d) 1 and 9
32. 32.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 18 (c) 9 (d) 10
33. 33.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
34. 34.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
35. 35.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)
36. 36.If a + b = 12 and ab = 32, what is a² + b²? (a) 208 (b) 80 (c) 144 (d) 112
37. 37.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 9025 (c) 7225 (d) 8075
38. 38.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 55 (c) 54 (d) 15
39. 39.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 529 (c) 361 (d) 193
40. 40.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 539 (d) 529

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