Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 487 · 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 28²? (a) 896 (b) 1024 (c) 784 (d) 794
2. 2.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
3. 3.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5939 (c) 6391 (d) 5929
4. 4.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 4)(x + 9) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
5. 5.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3721 (b) 3731 (c) 3601 (d) 3481
6. 6.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 2)(x + 8) (d) (x + 1)(x + 16)
7. 7.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)
8. 8.If a + b = 19 and ab = 88, what is (a − b)²? (a) 9 (b) 185 (c) 3 (d) 361
9. 9.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 63 (d) 62
10. 10.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
11. 11.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3025 (d) 3035
12. 12.If a + b = 11 and ab = 30, what is a² + b²? (a) 91 (b) 121 (c) 181 (d) 61
13. 13.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 8281 (c) 7921 (d) 8101
14. 14.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)
15. 15.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 3625 (b) 4225 (c) 4235 (d) 3025
16. 16.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
17. 17.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 16 (d) 4
18. 18.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 11 (d) 13
19. 19.What is the product of the roots of x² − 14x + 45 = 0? (a) 44 (b) 45 (c) 14 (d) 46
20. 20.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 6425 (c) 7225 (d) 5625
21. 21.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 1 (d) 289
22. 22.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 6) (d) (x + 1)(x + 10)
23. 23.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
24. 24.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 2 (c) 4 (d) 36
25. 25.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 256 (c) 366 (d) 201
26. 26.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1444 (c) 1454 (d) 1764
27. 27.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 3 (c) 9 (d) 121
28. 28.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x − 3)(x − 4) (c) (x + 1)(x + 12) (d) (x + 1)(x + 6)
29. 29.If a + b = 13 and ab = 12, what is a² + b²? (a) 193 (b) 169 (c) 157 (d) 145
30. 30.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 169 (c) 409 (d) 289
31. 31.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 4 (d) 7
32. 32.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 2 and 36 (d) 1 and 72
33. 33.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 541 (c) 361 (d) 181
34. 34.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3616 (c) 4096 (d) 3136
35. 35.If a + b = 10 and ab = 16, what is (a − b)²? (a) 36 (b) 100 (c) 6 (d) 68
36. 36.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
37. 37.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1774 (c) 1604 (d) 1444
38. 38.If a + b = 12 and ab = 11, what is (a − b)²? (a) 100 (b) 10 (c) 122 (d) 144
39. 39.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 396 (c) 324 (d) 484
40. 40.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 5635 (c) 6375 (d) 5625

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