Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 758 · 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.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2401 (b) 2499 (c) 2601 (d) 2411
2. 2.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 7 and 9 (d) 3 and 21
3. 3.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 4 (c) 64 (d) 2
4. 4.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 181 (c) 361 (d) 1
5. 5.What is the sum of the roots of x² − 11x + 18 = 0? (a) 11 (b) 12 (c) 10 (d) 18
6. 6.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 2035 (b) 1625 (c) 1225 (d) 2025
7. 7.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
8. 8.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 17 (d) 63
9. 9.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 884 (c) 1156 (d) 676
10. 10.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)
11. 11.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2401 (b) 2601 (c) 2611 (d) 2501
12. 12.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 9 (c) 225 (d) 3
13. 13.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 14 (d) 16
14. 14.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 20 (c) 36 (d) 28
15. 15.If a + b = 13 and ab = 42, what is a² + b²? (a) 253 (b) 169 (c) 85 (d) 127
16. 16.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 13 (c) 42 (d) 43
17. 17.If a + b = 16 and ab = 48, what is a² + b²? (a) 160 (b) 256 (c) 352 (d) 208
18. 18.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4624 (c) 5184 (d) 4904
19. 19.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 151 (c) 196 (d) 286
20. 20.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 324 (c) 180 (d) 252
21. 21.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)
22. 22.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 12 (d) 14
23. 23.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2926 (d) 2916
24. 24.If a + b = 11 and ab = 18, what is (a − b)²? (a) 49 (b) 85 (c) 7 (d) 121
25. 25.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 9 (c) 225 (d) 153
26. 26.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 121 (c) 73 (d) 97
27. 27.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 225 (d) 337
28. 28.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
29. 29.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 202 (c) 400 (d) 4
30. 30.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)
31. 31.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11035 (c) 9025 (d) 11025
32. 32.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1691 (d) 1681
33. 33.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 289 (c) 529 (d) 409
34. 34.What are the roots of x² − 9x + 14 = 0? (a) 3 and 6 (b) 2 and 7 (c) 1 and 14 (d) 1 and 8
35. 35.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
36. 36.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
37. 37.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)
38. 38.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9984 (b) 9216 (c) 10816 (d) 9226
39. 39.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 9 (c) 18 (d) 19
40. 40.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 139 (c) 109 (d) 229

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