Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 135 · 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 − 13x + 36 = 0? (a) 14 (b) 12 (c) 36 (d) 13
2. 2.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1604 (c) 1764 (d) 1444
3. 3.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 2 and 36 (d) 1 and 16
4. 4.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
5. 5.What are the roots of x² − 13x + 36 = 0? (a) 1 and 36 (b) 4 and 9 (c) 1 and 12 (d) 2 and 18
6. 6.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
7. 7.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1596 (c) 1444 (d) 1454
8. 8.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 9 (d) 7
9. 9.What is the sum of the roots of x² − 9x + 14 = 0? (a) 14 (b) 8 (c) 9 (d) 10
10. 10.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2809 (c) 2209 (d) 2491
11. 11.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
12. 12.If a + b = 20 and ab = 96, what is (a − b)²? (a) 400 (b) 16 (c) 4 (d) 208
13. 13.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)
14. 14.If a + b = 9 and ab = 8, what is a² + b²? (a) 65 (b) 97 (c) 81 (d) 73
15. 15.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 494 (c) 484 (d) 324
16. 16.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
17. 17.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 25 (c) 3 (d) 9
18. 18.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 9975 (d) 11025
19. 19.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
20. 20.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4634 (c) 4624 (d) 4896
21. 21.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 170 (c) 324 (d) 478
22. 22.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x − 4)(x − 9) (c) (x + 1)(x + 36) (d) (x + 1)(x + 12)
23. 23.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 160 (c) 64 (d) 8
24. 24.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 61 (c) 91 (d) 1
25. 25.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1156 (c) 676 (d) 1166
26. 26.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
27. 27.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 185 (c) 361 (d) 3
28. 28.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 361 (c) 271 (d) 181
29. 29.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 10 (c) 8 (d) 9
30. 30.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 9 (c) 10 (d) 20
31. 31.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
32. 32.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)
33. 33.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 7569 (c) 8649 (d) 8109
34. 34.If a + b = 13 and ab = 36, what is (a − b)²? (a) 97 (b) 169 (c) 5 (d) 25
35. 35.If a + b = 7 and ab = 12, what is a² + b²? (a) 49 (b) 73 (c) 25 (d) 37
36. 36.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1946 (c) 1936 (d) 1296
37. 37.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 196 (c) 148 (d) 172
38. 38.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 334 (c) 484 (d) 324
39. 39.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 9409 (b) 10009 (c) 10619 (d) 10609
40. 40.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 29 (c) 11 (d) 28

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