Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 208 · 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 = 19 and ab = 90, what is a² + b²? (a) 361 (b) 181 (c) 541 (d) 271
2. 2.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4875 (c) 4235 (d) 4225
3. 3.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 382 (c) 256 (d) 130
4. 4.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 2 (c) 4 (d) 100
5. 5.What is the sum of the roots of x² − 9x + 18 = 0? (a) 10 (b) 18 (c) 9 (d) 8
6. 6.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 122 (c) 100 (d) 144
7. 7.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
8. 8.What is the sum of the roots of x² − 10x + 24 = 0? (a) 9 (b) 10 (c) 11 (d) 24
9. 9.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4096 (b) 4106 (c) 3136 (d) 3616
10. 10.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 229 (c) 139 (d) 109
11. 11.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6409 (c) 6889 (d) 6899
12. 12.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4884 (c) 4366 (d) 5476
13. 13.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 7225 (d) 8125
14. 14.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6094 (b) 6724 (c) 6084 (d) 6396
15. 15.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 181 (d) 361
16. 16.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
17. 17.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
18. 18.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
19. 19.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 14 (c) 15 (d) 54
20. 20.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 468 (c) 324 (d) 252
21. 21.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 202 (d) 301
22. 22.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 63 (c) 16 (d) 62
23. 23.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5625 (b) 7225 (c) 6375 (d) 5635
24. 24.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 160 (c) 256 (d) 8
25. 25.If a + b = 15 and ab = 50, what is (a − b)²? (a) 5 (b) 225 (c) 125 (d) 25
26. 26.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
27. 27.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 5184 (c) 4896 (d) 4624
28. 28.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 21 (c) 20 (d) 9
29. 29.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 1 and 12 (c) 2 and 6 (d) 3 and 4
30. 30.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 72 (c) 1 and 16 (d) 2 and 36
31. 31.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 1)(x + 42) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
32. 32.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 7 (c) 9 (d) 1
33. 33.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 3)(x + 6) (c) (x − 3)(x − 6) (d) (x + 1)(x + 8)
34. 34.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 304 (c) 400 (d) 592
35. 35.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 739 (c) 729 (d) 891
36. 36.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
37. 37.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 16 (c) 4 (d) 80
38. 38.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 896 (c) 794 (d) 784
39. 39.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2209 (b) 2819 (c) 2509 (d) 2809
40. 40.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 8464 (c) 8474 (d) 7744

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