Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 484 · 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 = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 8 (c) 100 (d) 64
2. 2.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 5)(x + 7) (c) (x + 1)(x + 11) (d) (x − 5)(x − 7)
3. 3.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 39 (c) 13 (d) 40
4. 4.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 484 (c) 164 (d) 244
5. 5.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 441 (d) 3
6. 6.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 11 (c) 10 (d) 18
7. 7.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 494 (d) 484
8. 8.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 223 (c) 421 (d) 289
9. 9.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)
10. 10.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
11. 11.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1296 (c) 1936 (d) 1306
12. 12.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 35 (d) 11
13. 13.If a + b = 3 and ab = 2, what is (a − b)²? (a) 7 (b) 1 (c) 9 (d) 5
14. 14.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 6425 (c) 7235 (d) 7225
15. 15.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6724 (b) 6084 (c) 6404 (d) 6734
16. 16.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 13 (d) 12
17. 17.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 3 (c) 289 (d) 9
18. 18.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 54 (c) 15 (d) 14
19. 19.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 117 (c) 225 (d) 3
20. 20.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 4 (c) 100 (d) 2
21. 21.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 1156 (c) 676 (d) 884
22. 22.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 1 and 13 (c) 1 and 48 (d) 2 and 24
23. 23.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 6384 (c) 5776 (d) 5786
24. 24.What is the product of the roots of x² − 14x + 45 = 0? (a) 45 (b) 46 (c) 14 (d) 44
25. 25.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5476 (b) 4356 (c) 5486 (d) 4916
26. 26.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
27. 27.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 3625 (c) 4235 (d) 3025
28. 28.If a + b = 20 and ab = 96, what is (a − b)²? (a) 16 (b) 400 (c) 4 (d) 208
29. 29.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 10 (c) 30 (d) 11
30. 30.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 289 (c) 169 (d) 229
31. 31.If a + b = 11 and ab = 18, what is (a − b)²? (a) 121 (b) 7 (c) 85 (d) 49
32. 32.If a + b = 10 and ab = 16, what is a² + b²? (a) 68 (b) 84 (c) 132 (d) 100
33. 33.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 18 (c) 19 (d) 17
34. 34.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 14 (c) 56 (d) 15
35. 35.Using (a − b)² = a² − 2ab + b², what is 98²? (a) 10404 (b) 9614 (c) 9996 (d) 9604
36. 36.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 7 and 8 (d) 2 and 28
37. 37.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3969 (c) 3979 (d) 3609
38. 38.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x + 3)(x + 4)
39. 39.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6399 (b) 6251 (c) 6241 (d) 6561
40. 40.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8291 (b) 7921 (c) 8281 (d) 8101

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