Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 379 · 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 = 16 and ab = 48, what is a² + b²? (a) 352 (b) 160 (c) 256 (d) 208
2. 2.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4901 (b) 5051 (c) 4761 (d) 5041
3. 3.If a + b = 12 and ab = 32, what is a² + b²? (a) 112 (b) 144 (c) 80 (d) 208
4. 4.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 2025 (c) 1235 (d) 1225
5. 5.What are the roots of x² − 8x + 15 = 0? (a) 1 and 7 (b) 2 and 6 (c) 3 and 5 (d) 1 and 15
6. 6.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
7. 7.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 12 (d) 13
8. 8.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 14 (c) 15 (d) 16
9. 9.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 130 (c) 4 (d) 256
10. 10.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 4771 (c) 4761 (d) 5041
11. 11.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 193 (c) 361 (d) 5
12. 12.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 125 (c) 213 (d) 169
13. 13.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
14. 14.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 85 (c) 169 (d) 253
15. 15.If a + b = 10 and ab = 21, what is (a − b)²? (a) 100 (b) 16 (c) 58 (d) 4
16. 16.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
17. 17.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 63 (c) 64 (d) 62
18. 18.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
19. 19.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
20. 20.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 13 (d) 43
21. 21.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 7579 (c) 8649 (d) 7569
22. 22.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 133 (c) 241 (d) 97
23. 23.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 100 (c) 58 (d) 142
24. 24.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
25. 25.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 11 (c) 12 (d) 28
26. 26.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 15 (c) 13 (d) 48
27. 27.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 1225 (c) 2035 (d) 2025
28. 28.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 198 (c) 117 (d) 90
29. 29.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 25 (c) 13 (d) 1
30. 30.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7931 (c) 7921 (d) 8281
31. 31.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 5)(x + 8) (c) (x − 5)(x − 8) (d) (x + 1)(x + 12)
32. 32.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
33. 33.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 1 and 56 (d) 2 and 28
34. 34.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 144 (c) 4 (d) 74
35. 35.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 484 (c) 164 (d) 324
36. 36.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x + 1)(x + 32) (d) (x − 4)(x − 8)
37. 37.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 15 (c) 56 (d) 14
38. 38.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8281 (c) 7921 (d) 8101
39. 39.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1764 (c) 1596 (d) 1444
40. 40.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 404 (c) 494 (d) 324

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