Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 964 · 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 66²? (a) 4884 (b) 4356 (c) 4366 (d) 5476
2. 2.If a + b = 14 and ab = 40, what is a² + b²? (a) 116 (b) 276 (c) 156 (d) 196
3. 3.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)
4. 4.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1521 (c) 1599 (d) 1531
5. 5.What are the roots of x² − 9x + 18 = 0? (a) 1 and 18 (b) 2 and 9 (c) 3 and 6 (d) 1 and 8
6. 6.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 169 (d) 1
7. 7.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 6) (c) (x + 2)(x + 3) (d) (x + 1)(x + 4)
8. 8.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
9. 9.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 289 (c) 219 (d) 149
10. 10.If a + b = 13 and ab = 12, what is (a − b)²? (a) 169 (b) 121 (c) 11 (d) 145
11. 11.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2411 (c) 2401 (d) 2601
12. 12.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4875 (c) 4235 (d) 4225
13. 13.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 10 (c) 18 (d) 11
14. 14.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 289 (d) 145
15. 15.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 133 (c) 122 (d) 144
16. 16.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 2)(x + 6) (c) (x − 2)(x − 6) (d) (x + 1)(x + 12)
17. 17.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 425 (d) 625
18. 18.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
19. 19.If a + b = 12 and ab = 20, what is (a − b)²? (a) 144 (b) 8 (c) 64 (d) 104
20. 20.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1089 (c) 909 (d) 1099
21. 21.If a + b = 8 and ab = 7, what is (a − b)²? (a) 50 (b) 6 (c) 64 (d) 36
22. 22.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 1 and 16 (c) 8 and 9 (d) 2 and 36
23. 23.Which is the correct factorisation of x² + 11x + 30? (a) (x − 5)(x − 6) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x + 1)(x + 30)
24. 24.If a + b = 13 and ab = 36, what is (a − b)²? (a) 5 (b) 169 (c) 25 (d) 97
25. 25.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 304 (c) 208 (d) 592
26. 26.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 9 (d) 3
27. 27.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5929 (c) 5939 (d) 6889
28. 28.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 5041 (c) 4899 (d) 4761
29. 29.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 1)(x + 11)
30. 30.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 109 (c) 214 (d) 74
31. 31.If a + b = 19 and ab = 88, what is a² + b²? (a) 537 (b) 361 (c) 185 (d) 273
32. 32.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 13 (c) 11 (d) 12
33. 33.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 2 (c) 4 (d) 36
34. 34.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
35. 35.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 9) (c) (x + 1)(x + 16) (d) (x + 2)(x + 8)
36. 36.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 169 (c) 49 (d) 7
37. 37.What is the product of the roots of x² − 11x + 18 = 0? (a) 11 (b) 18 (c) 17 (d) 19
38. 38.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 576 (c) 256 (d) 416
39. 39.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
40. 40.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3249 (c) 3591 (d) 3259

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