Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 575 · 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.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
2. 2.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 18 (c) 9 (d) 17
3. 3.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)
4. 4.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
5. 5.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 729 (d) 1089
6. 6.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2304 (c) 2504 (d) 2704
7. 7.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 64 (c) 78 (d) 57
8. 8.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
9. 9.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 2)(x + 7) (c) (x − 2)(x − 7) (d) (x + 1)(x + 14)
10. 10.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
11. 11.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 10 (c) 7 (d) 8
12. 12.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 22 (c) 13 (d) 10
13. 13.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 7396 (c) 8846 (d) 8836
14. 14.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271
15. 15.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 9 (d) 149
16. 16.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
17. 17.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
18. 18.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 13 (c) 12 (d) 7
19. 19.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 68 (c) 132 (d) 84
20. 20.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 1)(x + 10) (d) (x + 3)(x + 8)
21. 21.If a + b = 17 and ab = 70, what is a² + b²? (a) 219 (b) 429 (c) 289 (d) 149
22. 22.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 1 and 24 (c) 2 and 12 (d) 4 and 6
23. 23.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 12 (c) 42 (d) 13
24. 24.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 4 (d) 58
25. 25.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
26. 26.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 9 (c) 121 (d) 65
27. 27.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 169 (c) 85 (d) 253
28. 28.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 18 (c) 16 (d) 72
29. 29.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 961 (d) 901
30. 30.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 6 and 7 (d) 1 and 42
31. 31.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 145 (c) 289 (d) 217
32. 32.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 10 (c) 12 (d) 11
33. 33.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
34. 34.If a + b = 12 and ab = 27, what is a² + b²? (a) 117 (b) 144 (c) 198 (d) 90
35. 35.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)
36. 36.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 5184 (c) 4624 (d) 4904
37. 37.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 180 (c) 252 (d) 468
38. 38.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6241 (c) 6571 (d) 6401
39. 39.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 5041 (d) 4761
40. 40.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10404 (b) 10414 (c) 9604 (d) 10004

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