Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 655 · 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.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
2. 2.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 5476 (c) 4366 (d) 4356
3. 3.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 7 and 8 (c) 1 and 56 (d) 2 and 28
4. 4.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 7569 (b) 8109 (c) 8659 (d) 8649
5. 5.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x + 1)(x + 8)
6. 6.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
7. 7.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1296 (c) 1584 (d) 1936
8. 8.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 49 (c) 3 (d) 9
9. 9.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 391 (b) 529 (c) 299 (d) 289
10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 14)
11. 11.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 537 (c) 361 (d) 273
12. 12.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 148 (c) 172 (d) 196
13. 13.If a + b = 11 and ab = 28, what is (a − b)²? (a) 3 (b) 9 (c) 121 (d) 65
14. 14.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 1 and 12 (c) 2 and 6 (d) 3 and 4
15. 15.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 25 (d) 23
16. 16.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11035 (c) 11025 (d) 10025
17. 17.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
18. 18.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1099 (c) 909 (d) 1089
19. 19.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 1 and 63 (d) 7 and 9
20. 20.If a + b = 16 and ab = 55, what is a² + b²? (a) 366 (b) 146 (c) 201 (d) 256
21. 21.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 62 (d) 63
22. 22.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 3 (c) 289 (d) 9
23. 23.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 15 (c) 54 (d) 16
24. 24.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 214 (c) 74 (d) 109
25. 25.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8464 (b) 7744 (c) 8474 (d) 8104
26. 26.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 384 (c) 266 (d) 256
27. 27.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 12 (c) 7 (d) 6
28. 28.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 20 (c) 9 (d) 21
29. 29.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 25 (c) 1 (d) 19
30. 30.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 9) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)
31. 31.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 9216 (c) 10016 (d) 10826
32. 32.Which is the correct factorisation of x² + 9x + 20? (a) (x − 4)(x − 5) (b) (x + 4)(x + 5) (c) (x + 1)(x + 8) (d) (x + 1)(x + 20)
33. 33.If a + b = 12 and ab = 35, what is (a − b)²? (a) 2 (b) 74 (c) 144 (d) 4
34. 34.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 13 (d) 14
35. 35.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)
36. 36.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 5 (c) 4 (d) 7
37. 37.If a + b = 13 and ab = 40, what is (a − b)²? (a) 3 (b) 9 (c) 169 (d) 89
38. 38.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 400 (c) 598 (d) 301
39. 39.What is the product of the roots of x² − 11x + 18 = 0? (a) 18 (b) 11 (c) 17 (d) 19
40. 40.Which is the correct factorisation of x² + 8x + 12? (a) (x + 2)(x + 6) (b) (x + 1)(x + 12) (c) (x − 2)(x − 6) (d) (x + 1)(x + 7)

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