Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 962 · 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 − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
2. 2.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
3. 3.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 16 (d) 14
4. 4.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
5. 5.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 1 and 7 (c) 3 and 5 (d) 2 and 6
6. 6.If a + b = 15 and ab = 50, what is a² + b²? (a) 225 (b) 175 (c) 125 (d) 325
7. 7.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6734 (c) 6724 (d) 6404
8. 8.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)
9. 9.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3364 (c) 3604 (d) 3854
10. 10.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 7579 (c) 7569 (d) 8649
11. 11.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 9 (c) 81 (d) 3
12. 12.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1764 (c) 1444 (d) 1774
13. 13.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 217 (d) 145
14. 14.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 16 (c) 2 and 8 (d) 1 and 9
15. 15.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 109 (c) 214 (d) 74
16. 16.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 8) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
17. 17.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 8 (c) 18 (d) 10
18. 18.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8281 (d) 8291
19. 19.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x − 2)(x − 9) (c) (x + 1)(x + 18) (d) (x + 1)(x + 10)
20. 20.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
21. 21.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 7 (c) 6 (d) 5
22. 22.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 55 (c) 15 (d) 53
23. 23.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7225 (c) 6425 (d) 7235
24. 24.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
25. 25.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 89 (c) 249 (d) 129
26. 26.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 12 (d) 14
27. 27.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 22 (c) 20 (d) 21
28. 28.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
29. 29.What are the roots of x² − 11x + 24 = 0? (a) 1 and 10 (b) 2 and 12 (c) 3 and 8 (d) 1 and 24
30. 30.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
31. 31.If a + b = 10 and ab = 21, what is (a − b)²? (a) 4 (b) 58 (c) 100 (d) 16
32. 32.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
33. 33.What is the product of the roots of x² − 10x + 24 = 0? (a) 24 (b) 10 (c) 23 (d) 25
34. 34.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 12 (c) 27 (d) 11
35. 35.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 12) (c) (x + 2)(x + 6) (d) (x + 1)(x + 7)
36. 36.What are the roots of x² − 13x + 36 = 0? (a) 4 and 9 (b) 1 and 12 (c) 1 and 36 (d) 2 and 18
37. 37.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
38. 38.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 2 and 28 (c) 1 and 56 (d) 7 and 8
39. 39.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 13 (c) 15 (d) 48
40. 40.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 484 (c) 334 (d) 324

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