Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 784 · 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 57²? (a) 3969 (b) 3259 (c) 3591 (d) 3249
2. 2.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 9025 (b) 8075 (c) 7235 (d) 7225
3. 3.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)
4. 4.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6241 (b) 6561 (c) 6571 (d) 6401
5. 5.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 925 (c) 1235 (d) 625
6. 6.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 729 (b) 891 (c) 739 (d) 1089
7. 7.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
8. 8.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 144 (c) 214 (d) 74
9. 9.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 425 (c) 635 (d) 625
10. 10.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2484 (b) 2126 (c) 2116 (d) 2916
11. 11.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 181 (d) 137
12. 12.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 301 (c) 400 (d) 202
13. 13.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
14. 14.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 2 and 15 (c) 1 and 10 (d) 5 and 6
15. 15.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 10 (c) 9 (d) 16
16. 16.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 271 (c) 181 (d) 361
17. 17.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
18. 18.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3374 (b) 3844 (c) 3596 (d) 3364
19. 19.If a + b = 9 and ab = 8, what is a² + b²? (a) 97 (b) 81 (c) 65 (d) 73
20. 20.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 14 (c) 15 (d) 13
21. 21.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 5329 (c) 4489 (d) 4909
22. 22.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 15 (d) 16
23. 23.If a + b = 12 and ab = 11, what is (a − b)²? (a) 122 (b) 10 (c) 100 (d) 144
24. 24.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 2 and 6 (c) 3 and 4 (d) 1 and 12
25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
26. 26.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 3)(x + 5) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
27. 27.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 27) (d) (x + 1)(x + 11)
28. 28.If a + b = 11 and ab = 10, what is (a − b)²? (a) 81 (b) 9 (c) 101 (d) 121
29. 29.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 1 and 11 (c) 4 and 8 (d) 1 and 32
30. 30.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4499 (c) 4891 (d) 4489
31. 31.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8836 (c) 8084 (d) 7396
32. 32.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 5 (c) 7 (d) 8
33. 33.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 10025 (b) 11035 (c) 9025 (d) 11025
34. 34.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 6)(x + 8) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
35. 35.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3969 (b) 3249 (c) 3979 (d) 3609
36. 36.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 16 (d) 18
37. 37.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 13 (c) 42 (d) 12
38. 38.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 484 (c) 494 (d) 404
39. 39.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 5625 (c) 4875 (d) 4235
40. 40.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 657 (d) 225

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