Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 349 · 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 51²? (a) 2611 (b) 2401 (c) 2601 (d) 2501
2. 2.If a + b = 5 and ab = 6, what is (a − b)²? (a) 1 (b) 19 (c) 13 (d) 25
3. 3.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
4. 4.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 635 (c) 875 (d) 625
5. 5.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 48 (c) 6 and 8 (d) 1 and 13
6. 6.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 5)(x + 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
7. 7.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1764 (c) 1596 (d) 1454
8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
9. 9.If a + b = 9 and ab = 20, what is (a − b)²? (a) 1 (b) 81 (c) 61 (d) 41
10. 10.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
11. 11.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 11 (c) 9 (d) 24
12. 12.If a + b = 11 and ab = 10, what is a² + b²? (a) 121 (b) 111 (c) 101 (d) 141
13. 13.If a + b = 16 and ab = 48, what is a² + b²? (a) 352 (b) 256 (c) 208 (d) 160
14. 14.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1024 (b) 1034 (c) 904 (d) 784
15. 15.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
16. 16.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
17. 17.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 2 and 27 (c) 1 and 14 (d) 6 and 9
18. 18.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 144 (c) 184 (d) 104
19. 19.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 17 (c) 71 (d) 72
20. 20.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
21. 21.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
22. 22.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 2 (c) 324 (d) 4
23. 23.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 19 (c) 25 (d) 13
24. 24.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
25. 25.If a + b = 13 and ab = 30, what is (a − b)²? (a) 109 (b) 49 (c) 169 (d) 7
26. 26.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 10 (c) 21 (d) 11
27. 27.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 1575 (d) 2025
28. 28.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 1099 (c) 909 (d) 729
29. 29.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 15 (d) 16
30. 30.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)
31. 31.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
32. 32.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9801 (c) 9811 (d) 9999
33. 33.If a + b = 13 and ab = 22, what is a² + b²? (a) 125 (b) 147 (c) 169 (d) 213
34. 34.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 899 (c) 961 (d) 851
35. 35.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 13 (c) 12 (d) 7
36. 36.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
37. 37.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 14 (c) 8 (d) 16
38. 38.If a + b = 9 and ab = 14, what is (a − b)²? (a) 25 (b) 81 (c) 53 (d) 5
39. 39.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8836 (b) 7396 (c) 8116 (d) 8846
40. 40.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)

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