Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 260 · 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.What is the product of the roots of x² − 14x + 45 = 0? (a) 45 (b) 14 (c) 46 (d) 44
2. 2.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x + 1)(x + 11) (d) (x − 5)(x − 7)
3. 3.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 1 and 14 (c) 6 and 9 (d) 2 and 27
4. 4.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 904 (c) 1024 (d) 1034
5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
6. 6.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 2 (c) 196 (d) 4
7. 7.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2025 (c) 1225 (d) 2035
8. 8.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 484 (c) 324 (d) 404
9. 9.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 55 (d) 53
10. 10.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1596 (c) 1764 (d) 1444
11. 11.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 79 (c) 100 (d) 58
12. 12.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 1 (c) 121 (d) 61
13. 13.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 49 (d) 47
14. 14.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 181 (d) 361
15. 15.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 12 (c) 8 (d) 7
16. 16.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 299 (b) 529 (c) 289 (d) 391
17. 17.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 202 (c) 301 (d) 400
18. 18.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 33 (c) 17 (d) 25
19. 19.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 13 (c) 12 (d) 11
20. 20.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1369 (c) 1379 (d) 1591
21. 21.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 56 (c) 16 (d) 15
22. 22.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6251 (c) 6561 (d) 6241
23. 23.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 169 (c) 97 (d) 73
24. 24.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 7225 (c) 9035 (d) 8125
25. 25.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2304 (c) 2704 (d) 2504
26. 26.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)
27. 27.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 841 (b) 899 (c) 851 (d) 961
28. 28.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7931 (c) 7921 (d) 8281
29. 29.What is the sum of the roots of x² − 11x + 28 = 0? (a) 12 (b) 11 (c) 28 (d) 10
30. 30.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3601 (c) 3481 (d) 3731
31. 31.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 89 (c) 9 (d) 3
32. 32.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 484 (c) 324 (d) 164
33. 33.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)
34. 34.What is the sum of the roots of x² − 10x + 16 = 0? (a) 16 (b) 11 (c) 9 (d) 10
35. 35.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 3)(x + 8) (c) (x − 3)(x − 8) (d) (x + 1)(x + 24)
36. 36.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 8 (c) 64 (d) 160
37. 37.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6401 (b) 6561 (c) 6571 (d) 6241
38. 38.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2126 (c) 2116 (d) 2916
39. 39.What is the product of the roots of x² − 9x + 20 = 0? (a) 9 (b) 21 (c) 19 (d) 20
40. 40.What are the roots of x² − 11x + 28 = 0? (a) 1 and 10 (b) 4 and 7 (c) 1 and 28 (d) 2 and 14

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