Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 762 · 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.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1235 (c) 1575 (d) 1225
2. 2.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 27 (c) 28 (d) 11
3. 3.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 7 and 8 (c) 1 and 56 (d) 1 and 14
4. 4.If a + b = 3 and ab = 2, what is a² + b²? (a) 7 (b) 5 (c) 9 (d) 13
5. 5.If a + b = 7 and ab = 6, what is (a − b)²? (a) 49 (b) 5 (c) 37 (d) 25
6. 6.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
7. 7.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 100 (c) 4 (d) 52
8. 8.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 5929 (c) 6899 (d) 6889
9. 9.If a + b = 14 and ab = 40, what is a² + b²? (a) 156 (b) 276 (c) 116 (d) 196
10. 10.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 20) (c) (x + 1)(x + 8) (d) (x − 4)(x − 5)
11. 11.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 30 (c) 29 (d) 31
12. 12.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 170 (c) 16 (d) 4
13. 13.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 657 (d) 225
14. 14.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 196 (c) 64 (d) 8
15. 15.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
16. 16.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 361 (d) 181
17. 17.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 7 (c) 65 (d) 49
18. 18.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 12 (c) 13 (d) 11
19. 19.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 11 (d) 12
20. 20.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1849 (b) 1369 (c) 1859 (d) 1609
21. 21.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6561 (c) 6571 (d) 6241
22. 22.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3584 (c) 3146 (d) 4096
23. 23.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 478 (c) 247 (d) 324
24. 24.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 14 (c) 2 and 27 (d) 1 and 54
25. 25.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 384 (c) 266 (d) 256
26. 26.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 91 (c) 181 (d) 61
27. 27.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 181 (d) 271
28. 28.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 324 (c) 2 (d) 4
29. 29.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 36 (c) 13 (d) 14
30. 30.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 1 and 18 (c) 2 and 9 (d) 1 and 8
31. 31.If a + b = 11 and ab = 18, what is (a − b)²? (a) 85 (b) 7 (c) 121 (d) 49
32. 32.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 4761 (c) 5051 (d) 5041
33. 33.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3601 (c) 3481 (d) 3731
34. 34.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 12 (d) 13
35. 35.What is the product of the roots of x² − 10x + 21 = 0? (a) 22 (b) 20 (c) 10 (d) 21
36. 36.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
37. 37.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
38. 38.If a + b = 17 and ab = 70, what is a² + b²? (a) 149 (b) 219 (c) 289 (d) 429
39. 39.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 12 (d) 14
40. 40.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 1156 (c) 884 (d) 686

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