Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 970 · 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.What is the product of the roots of x² − 13x + 40 = 0? (a) 39 (b) 13 (c) 40 (d) 41
2. 2.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
3. 3.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
4. 4.What are the roots of x² − 14x + 45 = 0? (a) 1 and 45 (b) 1 and 13 (c) 5 and 9 (d) 3 and 15
5. 5.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 2 and 36 (d) 1 and 72
6. 6.If a + b = 14 and ab = 40, what is (a − b)²? (a) 196 (b) 116 (c) 36 (d) 6
7. 7.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 9 (c) 16 (d) 11
8. 8.What is the product of the roots of x² − 7x + 10 = 0? (a) 9 (b) 11 (c) 7 (d) 10
9. 9.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1296 (c) 1306 (d) 1584
10. 10.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 12 (c) 28 (d) 11
11. 11.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
12. 12.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 361 (c) 1 (d) 271
13. 13.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8109 (c) 7569 (d) 8649
14. 14.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 4 (c) 324 (d) 2
15. 15.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3146 (b) 3584 (c) 3136 (d) 4096
16. 16.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3969 (b) 3259 (c) 3249 (d) 3591
17. 17.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 145 (c) 217 (d) 289
18. 18.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)
19. 19.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
20. 20.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 10025 (c) 11035 (d) 9025
21. 21.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 15 (c) 45 (d) 14
22. 22.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 6 and 7 (c) 2 and 21 (d) 1 and 42
23. 23.What is the product of the roots of x² − 11x + 28 = 0? (a) 29 (b) 11 (c) 28 (d) 27
24. 24.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 89 (c) 169 (d) 129
25. 25.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
26. 26.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 13 (c) 11 (d) 27
27. 27.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2116 (c) 2916 (d) 2484
28. 28.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
29. 29.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 202 (d) 2
30. 30.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 256 (c) 201 (d) 366
31. 31.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 11 (c) 1 and 32 (d) 4 and 8
32. 32.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
33. 33.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 32) (c) (x + 4)(x + 8) (d) (x + 1)(x + 11)
34. 34.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 52 (c) 100 (d) 2
35. 35.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 48 (c) 47 (d) 14
36. 36.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
37. 37.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 5476 (c) 4356 (d) 4366
38. 38.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 916 (c) 676 (d) 1166
39. 39.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
40. 40.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 12 (c) 13 (d) 7

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