Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 214 · 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.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 289 (c) 145 (d) 433
2. 2.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
3. 3.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 1 and 72 (d) 2 and 36
4. 4.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 46 (c) 44 (d) 45
5. 5.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 1)(x + 15)
6. 6.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 37 (c) 35 (d) 36
7. 7.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 8101 (c) 7921 (d) 8291
8. 8.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 1 and 12 (c) 2 and 21 (d) 6 and 7
9. 9.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 13 (d) 14
10. 10.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 2025 (c) 1225 (d) 1575
11. 11.What is the product of the roots of x² − 8x + 15 = 0? (a) 8 (b) 14 (c) 16 (d) 15
12. 12.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 163 (c) 262 (d) 130
13. 13.If a + b = 15 and ab = 56, what is a² + b²? (a) 337 (b) 225 (c) 169 (d) 113
14. 14.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
15. 15.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 127 (c) 1 (d) 85
16. 16.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 576 (c) 256 (d) 586
17. 17.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 185 (c) 9 (d) 361
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 + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
20. 20.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 529 (c) 289 (d) 409
21. 21.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 50 (c) 36 (d) 6
22. 22.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 5929 (c) 6409 (d) 6889
23. 23.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 61 (c) 1 (d) 121
24. 24.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1444 (c) 1764 (d) 1454
25. 25.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 7 and 9 (d) 1 and 63
26. 26.If a + b = 13 and ab = 22, what is (a − b)²? (a) 81 (b) 9 (c) 169 (d) 125
27. 27.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 400 (d) 202
28. 28.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10201 (b) 10211 (c) 9801 (d) 10001
29. 29.If a + b = 6 and ab = 5, what is (a − b)²? (a) 26 (b) 4 (c) 16 (d) 36
30. 30.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 28 (c) 26 (d) 27
31. 31.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 6 (c) 116 (d) 36
32. 32.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
33. 33.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 225 (c) 657 (d) 333
34. 34.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 9 (c) 8 (d) 7
35. 35.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 185 (c) 537 (d) 361
36. 36.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 1 (d) 217
37. 37.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 202 (d) 301
38. 38.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 42 (c) 41 (d) 13
39. 39.If a + b = 3 and ab = 2, what is a² + b²? (a) 5 (b) 9 (c) 13 (d) 7
40. 40.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)

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