Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 427 · 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 + 9x + 14? (a) (x + 1)(x + 14) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
2. 2.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1774 (b) 1444 (c) 1604 (d) 1764
3. 3.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 193 (c) 361 (d) 25
4. 4.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
5. 5.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10001 (c) 10201 (d) 10211
6. 6.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 10025 (c) 11025 (d) 9025
7. 7.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 11 (c) 1 and 27 (d) 3 and 9
8. 8.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
9. 9.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 73 (d) 72
10. 10.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 36 (c) 100 (d) 68
11. 11.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3364 (b) 3604 (c) 3854 (d) 3844
12. 12.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 247 (c) 478 (d) 170
13. 13.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 2 and 27 (c) 1 and 54 (d) 1 and 14
14. 14.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 1)(x + 18) (c) (x − 3)(x − 6) (d) (x + 3)(x + 6)
15. 15.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 304 (d) 400
16. 16.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
17. 17.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
18. 18.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)
19. 19.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 1 and 35 (c) 2 and 10 (d) 5 and 7
20. 20.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 149 (d) 9
21. 21.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 5 (c) 6 (d) 8
22. 22.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)
23. 23.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 13 (c) 11 (d) 12
24. 24.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 48 (d) 14
25. 25.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1225 (c) 1235 (d) 1575
26. 26.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 24 (c) 10 (d) 12
27. 27.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
28. 28.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 19 (c) 18 (d) 17
29. 29.If a + b = 6 and ab = 5, what is a² + b²? (a) 46 (b) 36 (c) 31 (d) 26
30. 30.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 729 (d) 1099
31. 31.If a + b = 14 and ab = 24, what is (a − b)²? (a) 100 (b) 10 (c) 148 (d) 196
32. 32.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 2 and 6 (c) 1 and 12 (d) 3 and 4
33. 33.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3609 (c) 3969 (d) 3979
34. 34.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 65 (c) 73 (d) 97
35. 35.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 299 (c) 391 (d) 529
36. 36.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 271 (d) 181
37. 37.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4761 (c) 4771 (d) 4899
38. 38.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 625 (b) 425 (c) 635 (d) 225
39. 39.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8464 (c) 7744 (d) 8096
40. 40.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6561 (c) 6241 (d) 6251

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