Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 884 · 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.If a + b = 14 and ab = 40, what is a² + b²? (a) 276 (b) 196 (c) 156 (d) 116
2. 2.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 26 (c) 31 (d) 46
3. 3.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
4. 4.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x + 1)(x + 40) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
5. 5.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 625 (c) 875 (d) 1225
6. 6.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
7. 7.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 7931 (c) 7921 (d) 8099
8. 8.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 529 (c) 539 (d) 289
9. 9.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 25 (c) 37 (d) 13
10. 10.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 175 (c) 325 (d) 125
11. 11.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2401 (c) 2601 (d) 2611
12. 12.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 45 (c) 81 (d) 3
13. 13.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1379 (c) 1369 (d) 1849
14. 14.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 289 (c) 223 (d) 421
15. 15.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 7225 (c) 8125 (d) 9025
16. 16.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2504 (b) 2704 (c) 2714 (d) 2304
17. 17.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
18. 18.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
19. 19.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 19 (d) 13
20. 20.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 11 (c) 12 (d) 10
21. 21.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 1 and 36 (c) 1 and 12 (d) 4 and 9
22. 22.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 181 (c) 1 (d) 361
23. 23.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 6 (c) 36 (d) 100
24. 24.What is the sum of the roots of x² − 10x + 16 = 0? (a) 9 (b) 16 (c) 10 (d) 11
25. 25.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 54 (c) 6 and 9 (d) 1 and 14
26. 26.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 576 (d) 586
27. 27.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 9614 (b) 10404 (c) 9996 (d) 9604
28. 28.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
29. 29.If a + b = 8 and ab = 12, what is a² + b²? (a) 40 (b) 52 (c) 64 (d) 88
30. 30.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 1235 (c) 1225 (d) 2025
31. 31.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 729 (c) 891 (d) 739
32. 32.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 85 (d) 127
33. 33.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 244 (c) 484 (d) 324
34. 34.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4884 (c) 4366 (d) 5476
35. 35.If a + b = 20 and ab = 96, what is a² + b²? (a) 304 (b) 208 (c) 592 (d) 400
36. 36.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6251 (c) 6399 (d) 6561
37. 37.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 16 (c) 4 (d) 10
38. 38.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 7 (c) 5 (d) 1
39. 39.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)
40. 40.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 56 (d) 16

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