Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 321 · 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 − 17x + 72 = 0? (a) 18 (b) 17 (c) 16 (d) 72
2. 2.If a + b = 14 and ab = 33, what is a² + b²? (a) 262 (b) 163 (c) 130 (d) 196
3. 3.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 169 (c) 229 (d) 289
4. 4.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
5. 5.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 5 (c) 7 (d) 6
6. 6.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 130 (c) 256 (d) 193
7. 7.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 361 (d) 181
8. 8.What is the sum of the roots of x² − 10x + 21 = 0? (a) 9 (b) 21 (c) 10 (d) 11
9. 9.What are the roots of x² − 5x + 6 = 0? (a) 1 and 6 (b) 1 and 4 (c) 2 and 3 (d) 2 and 4
10. 10.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 42 (d) 13
11. 11.What is the product of the roots of x² − 9x + 18 = 0? (a) 19 (b) 18 (c) 17 (d) 9
12. 12.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6409 (c) 5929 (d) 6889
13. 13.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 1)(x + 9) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
14. 14.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 4 (c) 2 (d) 256
15. 15.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 10 (d) 8
16. 16.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2484 (c) 2916 (d) 2116
17. 17.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 586 (c) 576 (d) 256
18. 18.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x − 4)(x − 9) (c) (x + 4)(x + 9) (d) (x + 1)(x + 36)
19. 19.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 25 (c) 157 (d) 289
20. 20.What is the sum of the roots of x² − 9x + 14 = 0? (a) 9 (b) 8 (c) 14 (d) 10
21. 21.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1599 (c) 1531 (d) 1521
22. 22.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 63 (c) 7 and 9 (d) 1 and 15
23. 23.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 686 (c) 1156 (d) 884
24. 24.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 14 (d) 16
25. 25.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 8 (c) 7 (d) 5
26. 26.What are the roots of x² − 13x + 40 = 0? (a) 1 and 12 (b) 2 and 20 (c) 5 and 8 (d) 1 and 40
27. 27.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 2)(x + 6)
28. 28.If a + b = 12 and ab = 11, what is (a − b)²? (a) 144 (b) 122 (c) 10 (d) 100
29. 29.If a + b = 8 and ab = 7, what is a² + b²? (a) 78 (b) 64 (c) 57 (d) 50
30. 30.If a + b = 10 and ab = 24, what is (a − b)²? (a) 52 (b) 4 (c) 2 (d) 100
31. 31.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 8075 (c) 7225 (d) 7235
32. 32.If a + b = 17 and ab = 72, what is (a − b)²? (a) 217 (b) 1 (c) 289 (d) 145
33. 33.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1601 (d) 1681
34. 34.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 10 (c) 11 (d) 24
35. 35.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 5 (c) 7 (d) 6
36. 36.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 37 (c) 19 (d) 13
37. 37.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 49 (c) 47 (d) 48
38. 38.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 11 (d) 35
39. 39.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 3 (c) 9 (d) 441
40. 40.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 14 (c) 1 and 54 (d) 6 and 9

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