Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 257 · 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 105²? (a) 11025 (b) 9025 (c) 10025 (d) 11035
2. 2.If a + b = 5 and ab = 6, what is a² + b²? (a) 13 (b) 19 (c) 25 (d) 37
3. 3.If a + b = 7 and ab = 12, what is (a − b)²? (a) 1 (b) 37 (c) 49 (d) 25
4. 4.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)
5. 5.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 130 (c) 4 (d) 256
6. 6.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 94 (c) 49 (d) 34
7. 7.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 1 and 48 (c) 2 and 24 (d) 6 and 8
8. 8.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 1 and 30 (d) 5 and 6
9. 9.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2819 (c) 2809 (d) 2509
10. 10.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1454 (c) 1444 (d) 1596
11. 11.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3584 (c) 3146 (d) 4096
12. 12.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8836 (c) 7406 (d) 8084
13. 13.If a + b = 7 and ab = 6, what is (a − b)²? (a) 25 (b) 37 (c) 49 (d) 5
14. 14.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)
15. 15.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6561 (b) 6401 (c) 6571 (d) 6241
16. 16.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 37 (c) 25 (d) 49
17. 17.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 144 (c) 90 (d) 117
18. 18.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 2 and 36 (c) 1 and 72 (d) 1 and 16
19. 19.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1936 (c) 1296 (d) 1946
20. 20.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 208 (c) 400 (d) 4
21. 21.What is the sum of the roots of x² − 10x + 21 = 0? (a) 10 (b) 21 (c) 11 (d) 9
22. 22.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 4 (c) 6 (d) 5
23. 23.If a + b = 12 and ab = 32, what is a² + b²? (a) 144 (b) 208 (c) 80 (d) 112
24. 24.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3025 (b) 3575 (c) 3035 (d) 4225
25. 25.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 11 (c) 13 (d) 27
26. 26.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 9 (c) 121 (d) 3
27. 27.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 3 (c) 9 (d) 29
28. 28.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 49 (c) 7 (d) 65
29. 29.If a + b = 9 and ab = 14, what is (a − b)²? (a) 25 (b) 5 (c) 53 (d) 81
30. 30.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 42 (d) 43
31. 31.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 8 (c) 5 (d) 6
32. 32.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
33. 33.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 4)(x + 6)
34. 34.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 10 (c) 23 (d) 24
35. 35.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 441 (d) 451
36. 36.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1575 (d) 1235
37. 37.If a + b = 13 and ab = 30, what is a² + b²? (a) 139 (b) 229 (c) 109 (d) 169
38. 38.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 676 (c) 884 (d) 1156
39. 39.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 5)(x + 9) (d) (x + 1)(x + 45)
40. 40.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 4 (c) 202 (d) 2

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