Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 159 · 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 93²? (a) 8659 (b) 7569 (c) 8109 (d) 8649
2. 2.If a + b = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 144 (c) 6 (d) 36
3. 3.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 541 (d) 181
4. 4.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 361 (c) 1 (d) 181
5. 5.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 256 (b) 576 (c) 384 (d) 266
6. 6.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 5 (c) 6 (d) 8
7. 7.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 202 (d) 400
8. 8.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)
9. 9.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1306 (c) 1584 (d) 1936
10. 10.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 8281 (c) 7921 (d) 8291
11. 11.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9216 (b) 10816 (c) 9984 (d) 9226
12. 12.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)
13. 13.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 14 (c) 15 (d) 45
14. 14.If a + b = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 100 (c) 10 (d) 148
15. 15.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 289 (c) 149 (d) 219
16. 16.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 2 (c) 130 (d) 4
17. 17.If a + b = 10 and ab = 21, what is a² + b²? (a) 58 (b) 79 (c) 100 (d) 142
18. 18.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 48 (d) 14
19. 19.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 14 (d) 54
20. 20.What is the product of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 53 (d) 55
21. 21.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)
22. 22.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 4761 (b) 5041 (c) 5051 (d) 4901
23. 23.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3035 (c) 3575 (d) 3025
24. 24.If a + b = 15 and ab = 44, what is a² + b²? (a) 313 (b) 225 (c) 181 (d) 137
25. 25.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 391 (c) 529 (d) 299
26. 26.What is the sum of the roots of x² − 16x + 63 = 0? (a) 63 (b) 17 (c) 16 (d) 15
27. 27.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3721 (c) 3731 (d) 3481
28. 28.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 371 (d) 361
29. 29.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
30. 30.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1936 (b) 1296 (c) 1946 (d) 1616
31. 31.If a + b = 6 and ab = 8, what is a² + b²? (a) 52 (b) 28 (c) 36 (d) 20
32. 32.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 3 (c) 225 (d) 441
33. 33.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1774 (c) 1604 (d) 1764
34. 34.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 1024 (b) 794 (c) 896 (d) 784
35. 35.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 13 (c) 48 (d) 14
36. 36.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
37. 37.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 1 and 11 (c) 2 and 10 (d) 3 and 9
38. 38.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 6 and 7 (c) 1 and 42 (d) 1 and 12
39. 39.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 14 (d) 13
40. 40.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 7225 (d) 6425

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