Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 871 · 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 + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 18) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)
2. 2.If a + b = 6 and ab = 5, what is a² + b²? (a) 26 (b) 36 (c) 46 (d) 31
3. 3.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6404 (c) 6734 (d) 6724
4. 4.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 5476 (d) 4884
5. 5.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 12) (b) (x − 2)(x − 6) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
6. 6.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 127 (c) 169 (d) 1
7. 7.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 324 (c) 6 (d) 180
8. 8.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 9 (c) 149 (d) 289
9. 9.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
11. 11.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 181 (d) 361
12. 12.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 1 and 13 (c) 1 and 45 (d) 3 and 15
13. 13.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 1 (c) 91 (d) 61
14. 14.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 1 and 12 (c) 6 and 7 (d) 2 and 21
15. 15.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 61 (c) 1 (d) 81
16. 16.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 64 (c) 57 (d) 50
17. 17.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2035 (c) 3025 (d) 2025
18. 18.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 361 (d) 537
19. 19.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3146 (c) 3584 (d) 3136
20. 20.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 81 (c) 45 (d) 9
21. 21.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 14 (d) 16
22. 22.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 1 and 10 (c) 1 and 24 (d) 2 and 12
23. 23.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 5 and 7 (c) 1 and 35 (d) 2 and 10
24. 24.If a + b = 13 and ab = 22, what is a² + b²? (a) 213 (b) 169 (c) 147 (d) 125
25. 25.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 9216 (c) 10826 (d) 10016
26. 26.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 28) (d) (x + 1)(x + 10)
27. 27.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 11 (c) 12 (d) 10
28. 28.What are the roots of x² − 9x + 14 = 0? (a) 1 and 8 (b) 1 and 14 (c) 2 and 7 (d) 3 and 6
29. 29.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 71 (d) 17
30. 30.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 1 and 48 (c) 6 and 8 (d) 2 and 24
31. 31.If a + b = 8 and ab = 15, what is (a − b)²? (a) 64 (b) 4 (c) 34 (d) 2
32. 32.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
33. 33.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 201 (c) 256 (d) 366
34. 34.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4499 (b) 4891 (c) 5329 (d) 4489
35. 35.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
36. 36.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 324 (c) 244 (d) 484
37. 37.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 24 (c) 10 (d) 25
38. 38.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 625 (d) 925
39. 39.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 13 (c) 25 (d) 1
40. 40.What is the product of the roots of x² − 12x + 27 = 0? (a) 26 (b) 12 (c) 28 (d) 27

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