Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 6 · 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.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2304 (c) 2704 (d) 2714
2. 2.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 36 (c) 6 (d) 196
3. 3.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
4. 4.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 400 (d) 4
5. 5.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
6. 6.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
7. 7.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 177 (c) 65 (d) 121
8. 8.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1454 (c) 1596 (d) 1764
9. 9.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4225 (c) 5635 (d) 4925
10. 10.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 19 (c) 21 (d) 20
11. 11.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7406 (c) 8836 (d) 7396
12. 12.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 81 (c) 109 (d) 67
13. 13.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 1089 (c) 739 (d) 891
14. 14.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
15. 15.If a + b = 13 and ab = 22, what is (a − b)²? (a) 169 (b) 125 (c) 9 (d) 81
16. 16.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 1 (c) 25 (d) 37
17. 17.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8116 (b) 7396 (c) 8846 (d) 8836
18. 18.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
19. 19.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 15 (c) 17 (d) 16
20. 20.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x − 5)(x − 8)
21. 21.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 104 (c) 8 (d) 64
22. 22.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9801 (c) 9811 (d) 9999
23. 23.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9409 (c) 9991 (d) 9419
24. 24.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 7 (c) 5 (d) 6
25. 25.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 2)(x + 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
26. 26.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
27. 27.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3604 (c) 3364 (d) 3854
28. 28.What are the roots of x² − 5x + 6 = 0? (a) 1 and 4 (b) 1 and 6 (c) 2 and 3 (d) 2 and 4
29. 29.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
30. 30.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 121 (c) 49 (d) 7
31. 31.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 4 (c) 100 (d) 52
32. 32.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 5476 (c) 4884 (d) 4366
33. 33.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 6 and 9 (c) 2 and 27 (d) 1 and 14
34. 34.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 400 (c) 301 (d) 598
35. 35.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5051 (c) 4901 (d) 5041
36. 36.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
37. 37.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
38. 38.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 7754 (c) 7744 (d) 8464
39. 39.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 875 (b) 635 (c) 1225 (d) 625
40. 40.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 5 (c) 7 (d) 6

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