Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 207 · 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.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
2. 2.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x − 3)(x − 6)
3. 3.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2601 (c) 2411 (d) 2401
4. 4.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 9 (c) 153 (d) 225
5. 5.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1599 (c) 1681 (d) 1531
6. 6.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 4 (c) 100 (d) 2
7. 7.If a + b = 12 and ab = 27, what is a² + b²? (a) 198 (b) 144 (c) 117 (d) 90
8. 8.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 289 (c) 409 (d) 229
9. 9.If a + b = 20 and ab = 99, what is (a − b)²? (a) 202 (b) 400 (c) 4 (d) 2
10. 10.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 10) (c) (x + 1)(x + 6) (d) (x + 2)(x + 5)
11. 11.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 6 (c) 5 (d) 7
12. 12.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)
13. 13.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 9 (c) 10 (d) 8
14. 14.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10816 (c) 10826 (d) 10016
15. 15.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 225 (c) 333 (d) 117
16. 16.If a + b = 7 and ab = 6, what is a² + b²? (a) 49 (b) 61 (c) 43 (d) 37
17. 17.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 4)(x + 9) (c) (x + 1)(x + 12) (d) (x − 4)(x − 9)
18. 18.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 3035 (c) 2525 (d) 2025
19. 19.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8281 (c) 8291 (d) 8101
20. 20.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 12 (c) 10 (d) 11
21. 21.What is the product of the roots of x² − 6x + 8 = 0? (a) 6 (b) 8 (c) 7 (d) 9
22. 22.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2116 (c) 2916 (d) 2126
23. 23.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 400 (c) 598 (d) 202
24. 24.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
25. 25.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 8 and 9 (d) 1 and 16
26. 26.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 7569 (c) 8091 (d) 8649
27. 27.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 11025 (d) 9975
28. 28.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1609 (c) 1849 (d) 1369
29. 29.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 15 (d) 13
30. 30.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 40 (c) 52 (d) 64
31. 31.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7744 (b) 8096 (c) 8464 (d) 7754
32. 32.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x + 1)(x + 12)
33. 33.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 4901 (c) 4761 (d) 5051
34. 34.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 196 (c) 256 (d) 376
35. 35.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10001 (c) 10211 (d) 10201
36. 36.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7225 (b) 5625 (c) 6425 (d) 7235
37. 37.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 100 (c) 52 (d) 2
38. 38.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x + 1)(x + 9) (d) (x − 4)(x − 6)
39. 39.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 896 (c) 784 (d) 794
40. 40.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 625 (d) 235

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