Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 311 · 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 = 10 and ab = 9, what is a² + b²? (a) 91 (b) 82 (c) 118 (d) 100
2. 2.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 961 (d) 901
3. 3.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2704 (c) 2504 (d) 2304
4. 4.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 1 and 13 (c) 2 and 24 (d) 1 and 48
5. 5.If a + b = 15 and ab = 50, what is (a − b)²? (a) 125 (b) 25 (c) 225 (d) 5
6. 6.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 225 (c) 441 (d) 3
7. 7.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 16 (c) 22 (d) 10
8. 8.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
9. 9.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3584 (c) 4096 (d) 3136
10. 10.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 30 (c) 29 (d) 11
11. 11.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 29 (c) 39 (d) 69
12. 12.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3844 (c) 3596 (d) 3374
13. 13.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 225 (c) 117 (d) 333
14. 14.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 27 (c) 2 and 10 (d) 1 and 11
15. 15.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)
16. 16.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 333 (c) 225 (d) 657
17. 17.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9226 (c) 9216 (d) 9984
18. 18.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 45) (c) (x + 1)(x + 13) (d) (x + 5)(x + 9)
19. 19.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 13 (d) 19
20. 20.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1235 (c) 625 (d) 1225
21. 21.What is the sum of the roots of x² − 14x + 48 = 0? (a) 13 (b) 48 (c) 15 (d) 14
22. 22.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 14 (c) 16 (d) 15
23. 23.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 1)(x + 7) (c) (x + 2)(x + 6) (d) (x + 1)(x + 12)
24. 24.If a + b = 11 and ab = 18, what is (a − b)²? (a) 7 (b) 121 (c) 49 (d) 85
25. 25.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 177 (d) 65
26. 26.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2484 (c) 2116 (d) 2916
27. 27.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 286 (c) 196 (d) 151
28. 28.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
29. 29.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6251 (c) 6561 (d) 6241
30. 30.If a + b = 13 and ab = 12, what is a² + b²? (a) 145 (b) 169 (c) 193 (d) 157
31. 31.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
32. 32.If a + b = 13 and ab = 12, what is (a − b)²? (a) 169 (b) 11 (c) 121 (d) 145
33. 33.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
34. 34.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
35. 35.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 253 (c) 85 (d) 169
36. 36.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 4)(x + 7)
37. 37.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 181 (d) 361
38. 38.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 63 (d) 15
39. 39.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 252 (c) 468 (d) 180
40. 40.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 18 (c) 8 (d) 9

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