Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 186 · 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 81²? (a) 6401 (b) 6561 (c) 6241 (d) 6571
2. 2.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x − 3)(x − 7) (c) (x + 1)(x + 9) (d) (x + 1)(x + 21)
3. 3.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)
4. 4.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
5. 5.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 15 (d) 16
6. 6.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 71 (c) 72 (d) 73
7. 7.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 686 (c) 1156 (d) 676
8. 8.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2411 (c) 2401 (d) 2601
9. 9.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 478 (c) 247 (d) 170
10. 10.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 121 (c) 81 (d) 41
11. 11.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 1)(x + 42) (d) (x + 6)(x + 7)
12. 12.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1575 (b) 2025 (c) 1235 (d) 1225
13. 13.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 21 (c) 33 (d) 17
14. 14.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x + 1)(x + 30) (d) (x − 5)(x − 6)
15. 15.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 42 (d) 13
16. 16.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 4771 (c) 4761 (d) 5041
17. 17.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)
18. 18.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 5 (c) 2 and 4 (d) 1 and 8
19. 19.What are the roots of x² − 11x + 28 = 0? (a) 2 and 14 (b) 4 and 7 (c) 1 and 28 (d) 1 and 10
20. 20.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4356 (c) 4366 (d) 5476
21. 21.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9991 (c) 9409 (d) 10609
22. 22.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)
23. 23.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5929 (b) 5939 (c) 6391 (d) 6889
24. 24.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3596 (b) 3374 (c) 3844 (d) 3364
25. 25.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10201 (c) 10211 (d) 10001
26. 26.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 13 (c) 35 (d) 11
27. 27.What is the sum of the roots of x² − 14x + 48 = 0? (a) 15 (b) 48 (c) 13 (d) 14
28. 28.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6251 (c) 6241 (d) 6561
29. 29.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
30. 30.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 84 (c) 100 (d) 132
31. 31.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3364 (c) 3604 (d) 3844
32. 32.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3731 (c) 3721 (d) 3601
33. 33.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 2 (c) 196 (d) 4
34. 34.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1099 (c) 1089 (d) 909
35. 35.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2219 (c) 2491 (d) 2809
36. 36.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 16 (c) 11 (d) 9
37. 37.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 1)(x + 8) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
38. 38.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
39. 39.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x − 5)(x − 7)
40. 40.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)

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