Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 457 · 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 sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 35 (c) 11 (d) 12
2. 2.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3364 (c) 3854 (d) 3844
3. 3.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x − 2)(x − 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
4. 4.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 7921 (d) 8281
5. 5.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 5625 (c) 4875 (d) 4225
6. 6.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 1034 (c) 1024 (d) 784
7. 7.If a + b = 3 and ab = 2, what is (a − b)²? (a) 1 (b) 7 (c) 5 (d) 9
8. 8.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)
9. 9.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5776 (c) 5786 (d) 7056
10. 10.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 3)(x + 9) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
11. 11.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1225 (c) 1235 (d) 1575
12. 12.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10619 (d) 10609
13. 13.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 9216 (c) 10826 (d) 10816
14. 14.If a + b = 17 and ab = 70, what is (a − b)²? (a) 9 (b) 289 (c) 149 (d) 3
15. 15.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 2 (d) 400
16. 16.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
17. 17.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6561 (b) 6251 (c) 6241 (d) 6399
18. 18.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 794 (c) 1024 (d) 784
19. 19.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1849 (c) 1369 (d) 1591
20. 20.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
21. 21.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 2 and 20 (c) 5 and 8 (d) 1 and 12
22. 22.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
23. 23.What are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 1 and 5 (c) 2 and 5 (d) 2 and 4
24. 24.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)
25. 25.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1681 (c) 1691 (d) 1521
26. 26.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 169 (c) 409 (d) 289
27. 27.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 1 and 14 (d) 2 and 28
28. 28.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1225 (d) 1235
29. 29.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 68 (c) 100 (d) 132
30. 30.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 3 (c) 9 (d) 25
31. 31.What are the roots of x² − 13x + 36 = 0? (a) 4 and 9 (b) 1 and 12 (c) 2 and 18 (d) 1 and 36
32. 32.If a + b = 5 and ab = 6, what is a² + b²? (a) 13 (b) 19 (c) 37 (d) 25
33. 33.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 2 and 21 (d) 6 and 7
34. 34.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x + 1)(x + 9)
35. 35.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 8 (c) 20 (d) 9
36. 36.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
37. 37.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 1 and 10 (d) 3 and 8
38. 38.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6724 (c) 6094 (d) 6396
39. 39.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 6 (c) 7 (d) 8
40. 40.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 8649 (c) 7569 (d) 8659

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