Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 840 · 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 = 11 and ab = 30, what is a² + b²? (a) 181 (b) 61 (c) 121 (d) 91
2. 2.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 3025 (c) 2525 (d) 2025
3. 3.If a + b = 15 and ab = 56, what is (a − b)²? (a) 1 (b) 113 (c) 225 (d) 169
4. 4.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 9801 (c) 10001 (d) 10201
5. 5.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 1 and 63 (d) 7 and 9
6. 6.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 153 (c) 225 (d) 189
7. 7.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6401 (c) 6561 (d) 6241
8. 8.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8474 (c) 7744 (d) 8464
9. 9.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8649 (b) 8091 (c) 7579 (d) 7569
10. 10.If a + b = 14 and ab = 45, what is a² + b²? (a) 151 (b) 286 (c) 106 (d) 196
11. 11.If a + b = 10 and ab = 16, what is a² + b²? (a) 84 (b) 132 (c) 100 (d) 68
12. 12.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
13. 13.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 145 (c) 433 (d) 289
14. 14.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 15 (c) 9 (d) 13
15. 15.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 1 and 11 (c) 2 and 10 (d) 3 and 9
16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
17. 17.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 8 (c) 5 (d) 7
18. 18.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 9 (d) 225
19. 19.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 11 (c) 12 (d) 10
20. 20.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)
21. 21.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2916 (c) 2116 (d) 2484
22. 22.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 484 (d) 494
23. 23.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 2 (c) 4 (d) 16
24. 24.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 484 (c) 334 (d) 396
25. 25.Using (a + b)² = a² + 2ab + b², what is 94²? (a) 7396 (b) 8836 (c) 8846 (d) 8116
26. 26.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 299 (c) 289 (d) 529
27. 27.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 1)(x + 40)
28. 28.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 16 (c) 1 and 72 (d) 8 and 9
29. 29.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
30. 30.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 5 (d) 4
31. 31.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 1)(x + 9) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
32. 32.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 164 (d) 244
33. 33.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 15 (d) 16
34. 34.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 7225 (c) 7235 (d) 8075
35. 35.If a + b = 3 and ab = 2, what is a² + b²? (a) 13 (b) 7 (c) 5 (d) 9
36. 36.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9975 (b) 9025 (c) 9035 (d) 11025
37. 37.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)
38. 38.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
39. 39.What are the roots of x² − 8x + 12 = 0? (a) 1 and 12 (b) 3 and 4 (c) 1 and 7 (d) 2 and 6
40. 40.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 2 and 8 (c) 3 and 7 (d) 1 and 9

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