Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 77 · 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 = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 52 (c) 4 (d) 2
2. 2.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
3. 3.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 14 (d) 48
4. 4.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x − 5)(x − 7) (c) (x + 1)(x + 11) (d) (x + 5)(x + 7)
5. 5.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)
6. 6.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 289 (c) 421 (d) 223
7. 7.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 169 (d) 225
8. 8.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 36 (c) 12 (d) 13
9. 9.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271
10. 10.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8101 (c) 8291 (d) 7921
11. 11.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 13 (c) 41 (d) 40
12. 12.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 9 (c) 185 (d) 361
13. 13.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 3 and 15 (c) 1 and 13 (d) 5 and 9
14. 14.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
15. 15.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 1 and 56 (c) 2 and 28 (d) 7 and 8
16. 16.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10211 (c) 10001 (d) 10201
17. 17.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 10 (c) 12 (d) 28
18. 18.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 382 (c) 193 (d) 130
19. 19.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 2525 (c) 3035 (d) 2025
20. 20.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 8125 (c) 7225 (d) 9035
21. 21.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10826 (c) 10816 (d) 10016
22. 22.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 214 (c) 74 (d) 144
23. 23.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 14 (d) 12
24. 24.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
25. 25.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 1 and 36 (c) 4 and 9 (d) 2 and 18
26. 26.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 4875 (d) 5625
27. 27.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 14 (c) 8 (d) 15
28. 28.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
29. 29.What is the product of the roots of x² − 11x + 30 = 0? (a) 30 (b) 29 (c) 31 (d) 11
30. 30.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1296 (c) 1946 (d) 1936
31. 31.If a + b = 15 and ab = 36, what is (a − b)²? (a) 225 (b) 9 (c) 81 (d) 153
32. 32.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 81 (c) 67 (d) 109
33. 33.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3854 (c) 3604 (d) 3844
34. 34.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
35. 35.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x + 1)(x + 12)
36. 36.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 103 (c) 85 (d) 157
37. 37.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x + 2)(x + 8)
38. 38.If a + b = 13 and ab = 40, what is a² + b²? (a) 129 (b) 249 (c) 89 (d) 169
39. 39.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x + 1)(x + 10)
40. 40.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 18 (c) 9 (d) 8

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