Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 881 · 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 + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
2. 2.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 4 and 6 (c) 1 and 24 (d) 2 and 12
3. 3.If a + b = 7 and ab = 10, what is a² + b²? (a) 29 (b) 39 (c) 69 (d) 49
4. 4.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 193 (c) 5 (d) 361
5. 5.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 2 and 6 (d) 1 and 6
6. 6.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 15 (c) 13 (d) 9
7. 7.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4909 (b) 5329 (c) 5339 (d) 4489
8. 8.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 11 (d) 12
9. 9.What is the product of the roots of x² − 14x + 45 = 0? (a) 45 (b) 44 (c) 46 (d) 14
10. 10.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 366 (c) 256 (d) 146
11. 11.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 15 (c) 8 (d) 7
12. 12.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 1)(x + 10)
13. 13.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
14. 14.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6571 (c) 6561 (d) 6401
15. 15.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 225 (c) 117 (d) 333
16. 16.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 196 (c) 106 (d) 286
17. 17.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 441 (d) 451
18. 18.What is the sum of the roots of x² − 12x + 32 = 0? (a) 12 (b) 11 (c) 13 (d) 32
19. 19.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
20. 20.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 4096 (b) 3584 (c) 3146 (d) 3136
21. 21.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)
22. 22.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3844 (c) 3374 (d) 3364
23. 23.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1625 (c) 2025 (d) 1225
24. 24.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 625 (c) 875 (d) 635
25. 25.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 127 (c) 169 (d) 253
26. 26.If a + b = 9 and ab = 18, what is a² + b²? (a) 45 (b) 81 (c) 63 (d) 117
27. 27.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
28. 28.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 299 (d) 391
29. 29.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2475 (c) 2025 (d) 3025
30. 30.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
31. 31.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 81 (c) 225 (d) 153
32. 32.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x + 1)(x + 12) (d) (x − 2)(x − 6)
33. 33.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1764 (c) 1444 (d) 1604
34. 34.What is the product of the roots of x² − 10x + 16 = 0? (a) 17 (b) 15 (c) 16 (d) 10
35. 35.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 58 (d) 4
36. 36.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
37. 37.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8836 (b) 7406 (c) 7396 (d) 8084
38. 38.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 2 (c) 256 (d) 4
39. 39.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
40. 40.If a + b = 10 and ab = 24, what is a² + b²? (a) 52 (b) 76 (c) 148 (d) 100

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