Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 219 · 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 = 8 and ab = 7, what is a² + b²? (a) 78 (b) 64 (c) 50 (d) 57
2. 2.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
3. 3.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 256 (c) 366 (d) 146
4. 4.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 33 (c) 17 (d) 25
5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
6. 6.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 289 (c) 145 (d) 433
7. 7.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4891 (c) 4489 (d) 4499
8. 8.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 1)(x + 18) (d) (x + 3)(x + 6)
9. 9.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x − 3)(x − 4) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
10. 10.If a + b = 6 and ab = 5, what is a² + b²? (a) 36 (b) 46 (c) 26 (d) 31
11. 11.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 625 (c) 225 (d) 235
12. 12.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 225 (d) 9
13. 13.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
14. 14.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 15 (d) 16
15. 15.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 144 (c) 4 (d) 16
16. 16.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
17. 17.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
18. 18.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2601 (c) 2401 (d) 2411
19. 19.If a + b = 14 and ab = 33, what is a² + b²? (a) 196 (b) 130 (c) 163 (d) 262
20. 20.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 1625 (d) 1225
21. 21.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 8109 (c) 8659 (d) 7569
22. 22.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
23. 23.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 181 (d) 541
24. 24.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
25. 25.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 41 (c) 81 (d) 121
26. 26.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1936 (c) 1306 (d) 1296
27. 27.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 16 (c) 4 (d) 400
28. 28.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 3 and 21 (d) 7 and 9
29. 29.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1296 (c) 1616 (d) 1936
30. 30.Which is the correct factorisation of x² + 10x + 21? (a) (x − 3)(x − 7) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
31. 31.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 286 (c) 196 (d) 151
32. 32.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 17 (c) 19 (d) 18
33. 33.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8104 (d) 8464
34. 34.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 7225 (b) 8125 (c) 9035 (d) 9025
35. 35.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 468 (c) 180 (d) 252
36. 36.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 35 (c) 36 (d) 12
37. 37.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 299 (c) 391 (d) 529
38. 38.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10001 (c) 10211 (d) 10201
39. 39.What is the sum of the roots of x² − 10x + 21 = 0? (a) 11 (b) 10 (c) 21 (d) 9
40. 40.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 1 (c) 41 (d) 81

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