Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 347 · 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 + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
2. 2.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4624 (b) 5194 (c) 5184 (d) 4904
3. 3.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 11 (c) 27 (d) 12
4. 4.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2116 (c) 2916 (d) 2516
5. 5.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4896 (c) 4624 (d) 4634
6. 6.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 37 (c) 13 (d) 25
7. 7.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 1034 (b) 1024 (c) 784 (d) 904
8. 8.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 14 (c) 1 and 56 (d) 7 and 8
9. 9.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 289 (c) 229 (d) 169
10. 10.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3584 (b) 3146 (c) 4096 (d) 3136
11. 11.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 57 (c) 50 (d) 78
12. 12.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2401 (c) 2601 (d) 2411
13. 13.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 127 (c) 169 (d) 85
14. 14.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
15. 15.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 6899 (c) 5929 (d) 6889
16. 16.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 13 (c) 42 (d) 41
17. 17.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)
18. 18.If a + b = 12 and ab = 11, what is a² + b²? (a) 122 (b) 166 (c) 144 (d) 133
19. 19.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)
20. 20.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 36) (c) (x + 1)(x + 12) (d) (x + 4)(x + 9)
21. 21.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 36 (c) 6 (d) 64
22. 22.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x − 4)(x − 7) (c) (x + 1)(x + 10) (d) (x + 4)(x + 7)
23. 23.If a + b = 7 and ab = 12, what is (a − b)²? (a) 49 (b) 1 (c) 37 (d) 25
24. 24.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 11 (c) 31 (d) 30
25. 25.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 4 (d) 5
26. 26.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
27. 27.If a + b = 6 and ab = 8, what is (a − b)²? (a) 36 (b) 20 (c) 2 (d) 4
28. 28.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1681 (d) 1691
29. 29.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 181 (d) 271
30. 30.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 6)
31. 31.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 49 (c) 37 (d) 25
32. 32.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9035 (c) 9025 (d) 9975
33. 33.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
34. 34.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 256 (c) 136 (d) 376
35. 35.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 97 (d) 121
36. 36.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 68 (c) 36 (d) 100
37. 37.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
38. 38.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7931 (c) 7921 (d) 8099
39. 39.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1849 (c) 1609 (d) 1369
40. 40.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 9984 (c) 9226 (d) 10816

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