Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 787 · 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 + 48 = 0? (a) 47 (b) 49 (c) 48 (d) 14
2. 2.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 28 (c) 11 (d) 12
3. 3.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
4. 4.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3025 (b) 3625 (c) 4235 (d) 4225
5. 5.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 289 (d) 1
6. 6.What are the roots of x² − 7x + 10 = 0? (a) 2 and 5 (b) 1 and 10 (c) 3 and 4 (d) 1 and 6
7. 7.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 46 (c) 26 (d) 36
8. 8.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 1 and 7 (c) 2 and 6 (d) 1 and 12
9. 9.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1521 (c) 1681 (d) 1531
10. 10.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 74 (c) 144 (d) 109
11. 11.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 14 (d) 15
12. 12.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
13. 13.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 7235 (c) 8075 (d) 9025
14. 14.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 160 (d) 352
15. 15.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)
16. 16.What is the sum of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 15 (d) 13
17. 17.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 81 (c) 153 (d) 225
18. 18.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 1024 (c) 904 (d) 1034
19. 19.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 625 (c) 1225 (d) 1235
20. 20.If a + b = 17 and ab = 60, what is a² + b²? (a) 289 (b) 169 (c) 229 (d) 409
21. 21.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
22. 22.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 225 (c) 125 (d) 25
23. 23.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)
24. 24.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
25. 25.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x − 2)(x − 7) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
26. 26.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
27. 27.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 113 (d) 1
28. 28.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 181 (d) 361
29. 29.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 41 (d) 13
30. 30.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8291 (c) 7921 (d) 8101
31. 31.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 25 (c) 193 (d) 361
32. 32.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
33. 33.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 18 (c) 3 and 6 (d) 1 and 8
34. 34.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 10 (c) 9 (d) 24
35. 35.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
36. 36.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
37. 37.If a + b = 18 and ab = 77, what is a² + b²? (a) 170 (b) 478 (c) 247 (d) 324
38. 38.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 25 (c) 13 (d) 19
39. 39.What are the roots of x² − 11x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 3 and 8 (d) 1 and 10
40. 40.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 49 (c) 7 (d) 169

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