Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 812 · 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 + 20) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
2. 2.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6409 (c) 5929 (d) 6889
3. 3.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
4. 4.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 9 (c) 10 (d) 14
5. 5.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2116 (c) 2916 (d) 2484
6. 6.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 9801 (d) 10001
7. 7.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1936 (c) 1296 (d) 1584
8. 8.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
9. 9.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 7 (c) 8 (d) 6
10. 10.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 1 and 48 (c) 2 and 24 (d) 6 and 8
11. 11.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 156 (c) 196 (d) 116
12. 12.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 35 (c) 13 (d) 12
13. 13.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 304 (d) 592
14. 14.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 15 (d) 1 and 63
15. 15.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 2 and 28 (d) 1 and 14
16. 16.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 5 (c) 25 (d) 169
17. 17.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 117 (c) 225 (d) 9
18. 18.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 1)(x + 36) (c) (x + 4)(x + 9) (d) (x − 4)(x − 9)
19. 19.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 31 (c) 11 (d) 30
20. 20.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 3 (c) 17 (d) 25
21. 21.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 12 (d) 14
22. 22.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 10009 (c) 9409 (d) 10619
23. 23.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3604 (c) 3844 (d) 3854
24. 24.If a + b = 21 and ab = 108, what is (a − b)²? (a) 3 (b) 9 (c) 441 (d) 225
25. 25.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9216 (c) 9226 (d) 10816
26. 26.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 193 (d) 361
27. 27.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 289 (c) 219 (d) 429
28. 28.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3609 (c) 3249 (d) 3979
29. 29.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 537 (c) 361 (d) 273
30. 30.What is the product of the roots of x² − 11x + 24 = 0? (a) 24 (b) 11 (c) 23 (d) 25
31. 31.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
32. 32.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 13 (c) 35 (d) 36
33. 33.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)
34. 34.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
35. 35.What is the product of the roots of x² − 7x + 12 = 0? (a) 12 (b) 13 (c) 7 (d) 11
36. 36.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 208 (c) 400 (d) 4
37. 37.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 43 (d) 41
38. 38.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 5 (d) 6
39. 39.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6561 (c) 6571 (d) 6401
40. 40.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x + 2)(x + 7) (d) (x − 2)(x − 7)

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