Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 304 · 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.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 15 (c) 13 (d) 14
2. 2.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2126 (c) 2116 (d) 2916
3. 3.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
4. 4.What is the sum of the roots of x² − 13x + 40 = 0? (a) 14 (b) 40 (c) 13 (d) 12
5. 5.If a + b = 9 and ab = 14, what is a² + b²? (a) 109 (b) 67 (c) 81 (d) 53
6. 6.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 9025 (c) 11025 (d) 11035
7. 7.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 10 (c) 1 and 6 (d) 2 and 5
8. 8.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
9. 9.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 8846 (c) 7396 (d) 8836
10. 10.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 4235 (c) 3625 (d) 4225
11. 11.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
12. 12.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)
13. 13.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 16 (d) 4
14. 14.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1936 (c) 1946 (d) 1296
15. 15.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 10 (c) 18 (d) 12
16. 16.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 1 (c) 9 (d) 7
17. 17.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
18. 18.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
19. 19.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 3 and 21 (d) 7 and 9
20. 20.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 3 and 15 (c) 1 and 45 (d) 5 and 9
21. 21.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 50 (c) 57 (d) 64
22. 22.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 49 (d) 47
23. 23.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
24. 24.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 13 (d) 11
25. 25.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 16 (d) 18
26. 26.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 15 (c) 14 (d) 56
27. 27.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1166 (b) 1156 (c) 916 (d) 676
28. 28.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 45 (c) 9 (d) 81
29. 29.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 193 (c) 382 (d) 256
30. 30.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 11 (c) 12 (d) 13
31. 31.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 13 (d) 12
32. 32.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 494 (d) 484
33. 33.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2411 (c) 2601 (d) 2401
34. 34.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6404 (c) 6084 (d) 6724
35. 35.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)
36. 36.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x − 4)(x − 8) (c) (x + 1)(x + 11) (d) (x + 4)(x + 8)
37. 37.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
38. 38.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10609 (c) 10619 (d) 9409
39. 39.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 72 (d) 73
40. 40.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 144 (c) 2 (d) 74

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