Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 7 · 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 + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x − 4)(x − 7) (d) (x + 1)(x + 28)
2. 2.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
3. 3.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
4. 4.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 3 (c) 9 (d) 225
5. 5.What is the product of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 33 (d) 31
6. 6.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
7. 7.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
8. 8.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 169 (c) 253 (d) 127
9. 9.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 375 (c) 625 (d) 225
10. 10.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
11. 11.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1936 (c) 1306 (d) 1584
12. 12.If a + b = 16 and ab = 48, what is (a − b)²? (a) 160 (b) 256 (c) 64 (d) 8
13. 13.What is the sum of the roots of x² − 7x + 10 = 0? (a) 7 (b) 6 (c) 8 (d) 10
14. 14.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 841 (c) 961 (d) 851
15. 15.If a + b = 16 and ab = 55, what is (a − b)²? (a) 256 (b) 36 (c) 146 (d) 6
16. 16.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 217 (d) 433
17. 17.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
18. 18.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 324 (d) 170
19. 19.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2484 (c) 2116 (d) 2916
20. 20.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 625 (d) 425
21. 21.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 63 (d) 62
22. 22.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2611 (c) 2601 (d) 2401
23. 23.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
24. 24.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 22 (c) 13 (d) 10
25. 25.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
26. 26.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2704 (c) 2714 (d) 2304
27. 27.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 441 (d) 451
28. 28.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 169 (c) 289 (d) 7
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 = 5 and ab = 6, what is a² + b²? (a) 19 (b) 13 (c) 37 (d) 25
31. 31.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 11 (c) 18 (d) 17
32. 32.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 325 (c) 225 (d) 125
33. 33.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 529 (c) 361 (d) 193
34. 34.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1024 (c) 784 (d) 1034
35. 35.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 324 (c) 244 (d) 484
36. 36.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 31 (c) 46 (d) 26
37. 37.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 1 and 63 (c) 3 and 21 (d) 1 and 15
38. 38.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 5 and 9 (c) 3 and 15 (d) 1 and 45
39. 39.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 4 (c) 16 (d) 208
40. 40.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 54 (d) 55

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