Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 500 · 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 86²? (a) 8836 (b) 7406 (c) 7396 (d) 8084
2. 2.If a + b = 10 and ab = 21, what is (a − b)²? (a) 16 (b) 58 (c) 100 (d) 4
3. 3.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x + 1)(x + 48)
4. 4.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 55 (c) 53 (d) 54
5. 5.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7225 (c) 7235 (d) 5625
6. 6.What are the roots of x² − 12x + 32 = 0? (a) 2 and 16 (b) 4 and 8 (c) 1 and 11 (d) 1 and 32
7. 7.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 225 (c) 333 (d) 657
8. 8.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x + 1)(x + 9)
9. 9.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 8 and 9 (d) 1 and 16
10. 10.If a + b = 14 and ab = 33, what is (a − b)²? (a) 130 (b) 196 (c) 64 (d) 8
11. 11.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1099 (c) 1089 (d) 909
12. 12.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3721 (d) 3601
13. 13.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1296 (b) 1584 (c) 1306 (d) 1936
14. 14.If a + b = 4 and ab = 3, what is a² + b²? (a) 13 (b) 22 (c) 16 (d) 10
15. 15.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 7)(x + 9)
16. 16.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 10) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
17. 17.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 8)
18. 18.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4235 (b) 4225 (c) 5625 (d) 4875
19. 19.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 6 and 8 (c) 1 and 13 (d) 1 and 48
20. 20.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 113 (c) 1 (d) 169
21. 21.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
22. 22.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 61 (c) 41 (d) 81
23. 23.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 5 (c) 13 (d) 7
24. 24.If a + b = 4 and ab = 3, what is (a − b)²? (a) 10 (b) 2 (c) 16 (d) 4
25. 25.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 2035 (c) 1625 (d) 2025
26. 26.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4356 (c) 4884 (d) 4366
27. 27.If a + b = 18 and ab = 80, what is (a − b)²? (a) 2 (b) 4 (c) 164 (d) 324
28. 28.If a + b = 20 and ab = 99, what is a² + b²? (a) 202 (b) 598 (c) 301 (d) 400
29. 29.If a + b = 12 and ab = 20, what is a² + b²? (a) 124 (b) 104 (c) 184 (d) 144
30. 30.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 875 (c) 635 (d) 625
31. 31.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 12 (c) 11 (d) 32
32. 32.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 84 (c) 100 (d) 132
33. 33.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 149 (c) 289 (d) 9
34. 34.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2509 (c) 2809 (d) 2209
35. 35.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3625 (c) 4225 (d) 3025
36. 36.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 441 (c) 399 (d) 361
37. 37.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 13 (c) 12 (d) 11
38. 38.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
39. 39.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 2 and 12 (c) 4 and 6 (d) 1 and 9
40. 40.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 3 (c) 81 (d) 9

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