Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 368 · 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.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 4 (c) 20 (d) 36
2. 2.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8291 (d) 8281
3. 3.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 1 and 42 (c) 2 and 21 (d) 6 and 7
4. 4.If a + b = 13 and ab = 30, what is a² + b²? (a) 229 (b) 139 (c) 109 (d) 169
5. 5.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 1 (c) 37 (d) 49
6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
7. 7.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 217 (d) 289
8. 8.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4235 (b) 3625 (c) 4225 (d) 3025
9. 9.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 2 (d) 202
10. 10.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 24 (c) 9 (d) 11
11. 11.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
12. 12.If a + b = 5 and ab = 4, what is (a − b)²? (a) 25 (b) 3 (c) 9 (d) 17
13. 13.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 9604 (c) 10414 (d) 10404
14. 14.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 441 (c) 3 (d) 225
15. 15.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 63 (d) 17
16. 16.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 1)(x + 16)
17. 17.If a + b = 11 and ab = 24, what is a² + b²? (a) 121 (b) 169 (c) 97 (d) 73
18. 18.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 14 (d) 13
19. 19.If a + b = 7 and ab = 6, what is a² + b²? (a) 37 (b) 49 (c) 61 (d) 43
20. 20.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2209 (c) 2809 (d) 2819
21. 21.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 729 (c) 1089 (d) 909
22. 22.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 145 (c) 193 (d) 157
23. 23.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)
24. 24.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1691 (c) 1601 (d) 1681
25. 25.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 244 (d) 164
26. 26.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x + 1)(x + 42) (d) (x − 6)(x − 7)
27. 27.What is the product of the roots of x² − 5x + 6 = 0? (a) 6 (b) 8 (c) 7 (d) 5
28. 28.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 4901 (c) 5041 (d) 4761
29. 29.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4884 (c) 5476 (d) 4356
30. 30.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 201 (c) 366 (d) 146
31. 31.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 14 (c) 36 (d) 13
32. 32.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
33. 33.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1936 (c) 1296 (d) 1306
34. 34.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)
35. 35.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 2)(x + 5) (c) (x + 1)(x + 6) (d) (x + 1)(x + 10)
36. 36.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 301 (d) 202
37. 37.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 184 (c) 104 (d) 124
38. 38.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1849 (c) 1369 (d) 1859
39. 39.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 399 (c) 361 (d) 441
40. 40.What is the sum of the roots of x² − 5x + 6 = 0? (a) 6 (b) 7 (c) 4 (d) 5

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