Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 621 · 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 39²? (a) 1599 (b) 1681 (c) 1521 (d) 1531
2. 2.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
3. 3.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)
4. 4.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 841 (c) 851 (d) 899
5. 5.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 1 and 14 (d) 2 and 28
6. 6.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)
7. 7.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 11 (c) 29 (d) 30
8. 8.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2704 (b) 2304 (c) 2496 (d) 2314
9. 9.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9801 (c) 9999 (d) 9811
10. 10.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 7 (c) 9 (d) 1
11. 11.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
12. 12.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 5)(x + 6) (d) (x + 1)(x + 10)
13. 13.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 334 (c) 396 (d) 484
14. 14.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 8 (c) 20 (d) 9
15. 15.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x − 6)(x − 8)
16. 16.Which is the correct factorisation of x² + 10x + 21? (a) (x + 3)(x + 7) (b) (x + 1)(x + 21) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
17. 17.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 5041 (c) 4901 (d) 4761
18. 18.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 1)(x + 13) (d) (x + 5)(x + 9)
19. 19.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x − 6)(x − 7)
20. 20.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 271 (c) 181 (d) 361
21. 21.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6401 (c) 6571 (d) 6561
22. 22.If a + b = 11 and ab = 30, what is a² + b²? (a) 181 (b) 61 (c) 121 (d) 91
23. 23.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 494 (d) 484
24. 24.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 7 and 9 (d) 3 and 21
25. 25.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1764 (b) 1444 (c) 1604 (d) 1774
26. 26.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 3 and 4 (c) 2 and 6 (d) 1 and 12
27. 27.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6094 (c) 6724 (d) 6084
28. 28.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 676 (b) 686 (c) 1156 (d) 884
29. 29.If a + b = 12 and ab = 20, what is (a − b)²? (a) 8 (b) 64 (c) 144 (d) 104
30. 30.If a + b = 15 and ab = 50, what is a² + b²? (a) 325 (b) 225 (c) 175 (d) 125
31. 31.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 16 (c) 170 (d) 4
32. 32.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4499 (c) 4489 (d) 4891
33. 33.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7396 (b) 8836 (c) 8084 (d) 7406
34. 34.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 169 (c) 289 (d) 409
35. 35.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 12 (c) 11 (d) 32
36. 36.If a + b = 6 and ab = 8, what is a² + b²? (a) 28 (b) 36 (c) 52 (d) 20
37. 37.What are the roots of x² − 12x + 32 = 0? (a) 1 and 11 (b) 1 and 32 (c) 4 and 8 (d) 2 and 16
38. 38.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 1)(x + 35)
39. 39.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 130 (c) 382 (d) 193
40. 40.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 586 (c) 416 (d) 576

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