Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 622 · 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.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
2. 2.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 180 (c) 468 (d) 252
3. 3.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8291 (c) 8281 (d) 7921
4. 4.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 2 (c) 400 (d) 4
5. 5.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 15 (c) 55 (d) 56
6. 6.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2504 (c) 2714 (d) 2704
7. 7.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 441 (d) 333
8. 8.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 12) (c) (x + 1)(x + 7) (d) (x + 2)(x + 6)
9. 9.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6889 (c) 6409 (d) 6899
10. 10.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
11. 11.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8649 (c) 7569 (d) 8109
12. 12.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 11 (d) 13
13. 13.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
14. 14.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8104 (d) 8464
15. 15.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 16) (c) (x + 1)(x + 9) (d) (x − 2)(x − 8)
16. 16.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 12 (c) 11 (d) 10
17. 17.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2704 (c) 2314 (d) 2496
18. 18.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 361 (c) 271 (d) 1
19. 19.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
20. 20.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 909 (c) 1089 (d) 729
21. 21.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10404 (d) 10414
22. 22.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
23. 23.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 30 (c) 2 and 15 (d) 1 and 10
24. 24.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 40 (c) 39 (d) 13
25. 25.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 1 (c) 13 (d) 25
26. 26.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 24 (c) 23 (d) 10
27. 27.What is the product of the roots of x² − 9x + 14 = 0? (a) 13 (b) 15 (c) 9 (d) 14
28. 28.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 9801 (c) 9999 (d) 10201
29. 29.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2126 (c) 2916 (d) 2484
30. 30.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 14 (c) 13 (d) 42
31. 31.What are the roots of x² − 13x + 42 = 0? (a) 6 and 7 (b) 1 and 42 (c) 1 and 12 (d) 2 and 21
32. 32.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
33. 33.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 8846 (b) 7396 (c) 8836 (d) 8116
34. 34.If a + b = 8 and ab = 12, what is (a − b)²? (a) 16 (b) 64 (c) 4 (d) 40
35. 35.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 72 (d) 1 and 16
36. 36.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)
37. 37.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 324 (c) 164 (d) 4
38. 38.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 45 (c) 46 (d) 14
39. 39.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 4 (c) 36 (d) 16
40. 40.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 181 (c) 271 (d) 361

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