Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 210 · 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 = 20 and ab = 99, what is a² + b²? (a) 400 (b) 301 (c) 202 (d) 598
2. 2.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 26 (c) 28 (d) 27
3. 3.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1444 (c) 1764 (d) 1454
4. 4.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2926 (c) 2116 (d) 2516
5. 5.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
6. 6.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 145 (c) 217 (d) 1
7. 7.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 1)(x + 13) (d) (x + 6)(x + 8)
8. 8.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)
9. 9.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 5635 (c) 4225 (d) 4925
10. 10.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 376 (c) 136 (d) 196
11. 11.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 7 (c) 49 (d) 289
12. 12.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 3 and 15 (c) 5 and 9 (d) 1 and 45
13. 13.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
14. 14.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 58 (c) 142 (d) 100
15. 15.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 160 (c) 208 (d) 352
16. 16.If a + b = 15 and ab = 44, what is a² + b²? (a) 181 (b) 225 (c) 137 (d) 313
17. 17.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 1)(x + 14) (c) (x + 1)(x + 8) (d) (x + 2)(x + 7)
18. 18.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 361 (c) 401 (d) 441
19. 19.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1936 (c) 1306 (d) 1584
20. 20.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3146 (c) 4096 (d) 3584
21. 21.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1849 (c) 1369 (d) 1379
22. 22.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6724 (c) 6734 (d) 6404
23. 23.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2401 (c) 2411 (d) 2601
24. 24.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
25. 25.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 8 (d) 9
26. 26.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 9 (c) 11 (d) 10
27. 27.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 185 (c) 3 (d) 361
28. 28.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 5)(x + 9)
29. 29.If a + b = 21 and ab = 108, what is a² + b²? (a) 657 (b) 225 (c) 441 (d) 333
30. 30.What are the roots of x² − 8x + 15 = 0? (a) 2 and 6 (b) 3 and 5 (c) 1 and 7 (d) 1 and 15
31. 31.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 63 (c) 16 (d) 62
32. 32.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 1 (c) 85 (d) 169
33. 33.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x − 3)(x − 7) (d) (x + 1)(x + 21)
34. 34.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 1 and 16 (d) 2 and 36
35. 35.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 36 (c) 64 (d) 6
36. 36.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 1 and 30 (c) 5 and 6 (d) 2 and 15
37. 37.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 2 and 14 (c) 1 and 10 (d) 4 and 7
38. 38.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 49 (c) 34 (d) 94
39. 39.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3035 (c) 3025 (d) 2025
40. 40.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)

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