Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 557 · 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 + 18) (c) (x − 3)(x − 6) (d) (x + 1)(x + 8)
2. 2.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 371 (c) 399 (d) 441
3. 3.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 592 (c) 208 (d) 400
4. 4.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)
5. 5.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 6 (c) 9 (d) 8
6. 6.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
7. 7.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 1)(x + 30)
8. 8.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 17 (d) 63
9. 9.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 324 (c) 247 (d) 170
10. 10.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 54 (c) 15 (d) 55
11. 11.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
12. 12.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 5)(x + 9)
13. 13.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 5 and 9 (c) 1 and 13 (d) 3 and 15
14. 14.What are the roots of x² − 9x + 20 = 0? (a) 1 and 20 (b) 1 and 8 (c) 2 and 10 (d) 4 and 5
15. 15.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2504 (c) 2704 (d) 2714
16. 16.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 48 (c) 14 (d) 15
17. 17.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 49 (d) 137
18. 18.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 256 (c) 201 (d) 366
19. 19.What are the roots of x² − 11x + 24 = 0? (a) 3 and 8 (b) 2 and 12 (c) 1 and 24 (d) 1 and 10
20. 20.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x + 1)(x + 12) (d) (x − 5)(x − 8)
21. 21.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 334 (c) 396 (d) 324
22. 22.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9035 (c) 11025 (d) 9025
23. 23.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2611 (c) 2401 (d) 2501
24. 24.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2035 (c) 2475 (d) 2025
25. 25.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576
26. 26.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 6391 (c) 5929 (d) 5939
27. 27.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 256 (d) 130
28. 28.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 625 (c) 925 (d) 1235
29. 29.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
30. 30.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 225 (c) 113 (d) 337
31. 31.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 57 (d) 55
32. 32.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 208 (c) 16 (d) 4
33. 33.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3604 (c) 3854 (d) 3364
34. 34.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 56 (d) 1 and 14
35. 35.What is the product of the roots of x² − 16x + 63 = 0? (a) 62 (b) 16 (c) 63 (d) 64
36. 36.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
37. 37.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 244 (c) 196 (d) 148
38. 38.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
39. 39.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 124 (c) 104 (d) 184
40. 40.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 5625 (c) 4235 (d) 4225

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