Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 276 · 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 + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
2. 2.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7921 (b) 8281 (c) 8099 (d) 7931
3. 3.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 1089 (c) 739 (d) 729
4. 4.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 333 (d) 225
5. 5.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5939 (c) 5929 (d) 6391
6. 6.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 113 (c) 225 (d) 169
7. 7.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 256 (c) 146 (d) 36
8. 8.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 10) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
9. 9.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 56 (c) 1 and 14 (d) 7 and 8
10. 10.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
11. 11.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
12. 12.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 196 (c) 163 (d) 130
13. 13.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
14. 14.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
15. 15.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2704 (c) 2314 (d) 2496
16. 16.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6094 (c) 6084 (d) 6396
17. 17.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 289 (c) 157 (d) 25
18. 18.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x − 4)(x − 9) (c) (x + 1)(x + 36) (d) (x + 1)(x + 12)
19. 19.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 884 (c) 686 (d) 676
20. 20.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
21. 21.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 3 (c) 9 (d) 29
22. 22.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
23. 23.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4489 (c) 4909 (d) 5329
24. 24.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 1 and 14 (c) 1 and 8 (d) 3 and 6
25. 25.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 576 (b) 256 (c) 586 (d) 416
26. 26.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 61 (c) 121 (d) 41
27. 27.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2475 (c) 3025 (d) 2025
28. 28.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6399 (c) 6251 (d) 6561
29. 29.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 1 and 8 (c) 2 and 10 (d) 1 and 20
30. 30.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
31. 31.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6241 (c) 6571 (d) 6561
32. 32.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 49 (c) 34 (d) 94
33. 33.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 5 and 7 (c) 1 and 11 (d) 1 and 35
34. 34.What is the sum of the roots of x² − 7x + 12 = 0? (a) 6 (b) 12 (c) 7 (d) 8
35. 35.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 16 (c) 10 (d) 15
36. 36.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2401 (c) 2601 (d) 2499
37. 37.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 175 (c) 125 (d) 225
38. 38.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2704 (c) 2304 (d) 2504
39. 39.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3721 (c) 3491 (d) 3481
40. 40.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)

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