Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 348 · 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 + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
2. 2.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
3. 3.What are the roots of x² − 7x + 10 = 0? (a) 3 and 4 (b) 1 and 10 (c) 2 and 5 (d) 1 and 6
4. 4.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 904 (c) 1034 (d) 1024
5. 5.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 225 (d) 169
6. 6.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
7. 7.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8836 (c) 8084 (d) 7396
8. 8.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 4 (c) 324 (d) 16
9. 9.What is the product of the roots of x² − 14x + 45 = 0? (a) 45 (b) 44 (c) 46 (d) 14
10. 10.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 19 (c) 18 (d) 17
11. 11.What are the roots of x² − 10x + 21 = 0? (a) 3 and 7 (b) 1 and 21 (c) 1 and 9 (d) 2 and 8
12. 12.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
13. 13.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 32 (c) 31 (d) 33
14. 14.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
15. 15.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 36 (d) 35
16. 16.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 441 (c) 333 (d) 225
17. 17.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
18. 18.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2304 (d) 2496
19. 19.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 125 (c) 175 (d) 325
20. 20.If a + b = 14 and ab = 45, what is a² + b²? (a) 196 (b) 286 (c) 106 (d) 151
21. 21.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 1024 (c) 784 (d) 794
22. 22.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
23. 23.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 169 (c) 229 (d) 289
24. 24.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 451 (c) 361 (d) 441
25. 25.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2126 (c) 2916 (d) 2484
26. 26.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
27. 27.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 10 (c) 4 (d) 2
28. 28.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
29. 29.What is the sum of the roots of x² − 9x + 20 = 0? (a) 20 (b) 8 (c) 10 (d) 9
30. 30.If a + b = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 82 (c) 64 (d) 8
31. 31.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 35 (c) 11 (d) 12
32. 32.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 19 (c) 13 (d) 25
33. 33.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 81 (c) 9 (d) 45
34. 34.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 3025 (d) 3625
35. 35.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 4909 (c) 5339 (d) 5329
36. 36.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
37. 37.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 1)(x + 9) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
38. 38.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 85 (c) 169 (d) 127
39. 39.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 16 (c) 4 (d) 400
40. 40.If a + b = 6 and ab = 8, what is a² + b²? (a) 20 (b) 28 (c) 52 (d) 36

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