Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 237 · 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 are the roots of x² − 6x + 8 = 0? (a) 1 and 8 (b) 1 and 5 (c) 2 and 5 (d) 2 and 4
2. 2.Which is the correct factorisation of x² + 8x + 12? (a) (x − 2)(x − 6) (b) (x + 2)(x + 6) (c) (x + 1)(x + 7) (d) (x + 1)(x + 12)
3. 3.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 26 (c) 28 (d) 12
4. 4.If a + b = 5 and ab = 4, what is a² + b²? (a) 21 (b) 17 (c) 25 (d) 33
5. 5.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 901 (c) 841 (d) 961
6. 6.Which is the correct factorisation of x² + 11x + 24? (a) (x + 3)(x + 8) (b) (x − 3)(x − 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
7. 7.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 78 (c) 64 (d) 57
8. 8.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 100 (c) 132 (d) 84
9. 9.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 537 (c) 273 (d) 185
10. 10.What is the product of the roots of x² − 14x + 45 = 0? (a) 14 (b) 45 (c) 46 (d) 44
11. 11.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
12. 12.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 2 and 28 (d) 1 and 14
13. 13.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 4 (c) 2 (d) 130
14. 14.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 10009 (c) 10609 (d) 9409
15. 15.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 169 (c) 127 (d) 1
16. 16.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 121 (c) 61 (d) 1
17. 17.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7225 (b) 6425 (c) 5625 (d) 7235
18. 18.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
19. 19.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 13 (d) 14
20. 20.If a + b = 16 and ab = 63, what is a² + b²? (a) 382 (b) 193 (c) 130 (d) 256
21. 21.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 1089 (c) 909 (d) 729
22. 22.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)
23. 23.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 49 (c) 1 (d) 37
24. 24.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 361 (c) 399 (d) 371
25. 25.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 14 (d) 15
26. 26.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 6409 (c) 5929 (d) 6899
27. 27.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
28. 28.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 35 (d) 11
29. 29.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 3)(x + 7) (c) (x + 1)(x + 21) (d) (x − 3)(x − 7)
30. 30.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
31. 31.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2126 (c) 2116 (d) 2916
32. 32.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)
33. 33.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 91 (c) 100 (d) 118
34. 34.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 12 (c) 42 (d) 13
35. 35.If a + b = 15 and ab = 50, what is a² + b²? (a) 125 (b) 175 (c) 225 (d) 325
36. 36.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 21 (c) 9 (d) 20
37. 37.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1584 (c) 1306 (d) 1936
38. 38.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3969 (c) 3979 (d) 3249
39. 39.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)
40. 40.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 10)

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