Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 543 · 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 = 10 and ab = 16, what is (a − b)²? (a) 36 (b) 6 (c) 68 (d) 100
2. 2.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
3. 3.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 1 and 12 (c) 3 and 4 (d) 2 and 6
4. 4.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
5. 5.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 3025 (c) 4225 (d) 4235
6. 6.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1379 (c) 1591 (d) 1849
7. 7.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 9409 (c) 9419 (d) 10609
8. 8.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 71 (d) 73
9. 9.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)
10. 10.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 576 (c) 586 (d) 416
11. 11.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
12. 12.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 686 (c) 1156 (d) 884
13. 13.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 6)
14. 14.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 2 and 6 (c) 3 and 4 (d) 1 and 12
15. 15.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 10 (c) 20 (d) 9
16. 16.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
17. 17.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 451 (c) 401 (d) 441
18. 18.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 43 (d) 41
19. 19.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 5625 (d) 4875
20. 20.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 19 (c) 17 (d) 18
21. 21.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
22. 22.If a + b = 14 and ab = 33, what is (a − b)²? (a) 8 (b) 130 (c) 64 (d) 196
23. 23.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 109 (c) 229 (d) 169
24. 24.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 11 (c) 12 (d) 13
25. 25.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4884 (c) 4366 (d) 5476
26. 26.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 97 (d) 121
27. 27.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 1)(x + 18)
28. 28.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 425 (d) 625
29. 29.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 144 (c) 214 (d) 109
30. 30.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 80 (c) 208 (d) 112
31. 31.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 3)(x + 5) (d) (x + 1)(x + 15)
32. 32.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1296 (c) 1946 (d) 1936
33. 33.If a + b = 17 and ab = 72, what is (a − b)²? (a) 1 (b) 217 (c) 289 (d) 145
34. 34.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2926 (d) 2916
35. 35.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1156 (c) 1166 (d) 676
36. 36.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 8464 (c) 7744 (d) 7754
37. 37.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 50 (d) 57
38. 38.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
39. 39.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 1089 (c) 729 (d) 891
40. 40.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 400 (d) 202

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