Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 359 · 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 = 11 and ab = 30, what is a² + b²? (a) 181 (b) 121 (c) 61 (d) 91
2. 2.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
3. 3.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1946 (c) 1616 (d) 1936
4. 4.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4356 (c) 5476 (d) 4366
5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x − 7)(x − 9)
6. 6.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 2025 (d) 3035
7. 7.If a + b = 8 and ab = 12, what is (a − b)²? (a) 16 (b) 40 (c) 64 (d) 4
8. 8.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 2 (c) 20 (d) 36
9. 9.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3601 (d) 3721
10. 10.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1099 (c) 909 (d) 1089
11. 11.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 16) (b) (x + 2)(x + 8) (c) (x + 1)(x + 9) (d) (x − 2)(x − 8)
12. 12.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
13. 13.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 3 and 4 (c) 2 and 5 (d) 1 and 6
14. 14.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 62 (d) 63
15. 15.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
16. 16.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
17. 17.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 361 (c) 541 (d) 271
18. 18.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4875 (b) 4235 (c) 4225 (d) 5625
19. 19.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)
20. 20.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 494 (d) 484
21. 21.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 1 and 24 (c) 2 and 12 (d) 4 and 6
22. 22.Using (a + b)² = a² + 2ab + b², what is 65²? (a) 4225 (b) 4235 (c) 3625 (d) 3025
23. 23.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 433 (c) 217 (d) 145
24. 24.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 401 (c) 451 (d) 441
25. 25.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
26. 26.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 6416 (c) 7066 (d) 7056
27. 27.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 47 (d) 49
28. 28.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3609 (b) 3249 (c) 3969 (d) 3979
29. 29.What is the sum of the roots of x² − 7x + 10 = 0? (a) 6 (b) 10 (c) 8 (d) 7
30. 30.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 16 (d) 1 and 72
31. 31.What is the product of the roots of x² − 9x + 20 = 0? (a) 20 (b) 21 (c) 9 (d) 19
32. 32.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 130 (c) 4 (d) 256
33. 33.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 11 (c) 10 (d) 12
34. 34.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 7569 (c) 7579 (d) 8649
35. 35.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 3 (c) 29 (d) 9
36. 36.What is the sum of the roots of x² − 12x + 27 = 0? (a) 27 (b) 11 (c) 12 (d) 13
37. 37.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 8 (c) 9 (d) 6
38. 38.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 337 (c) 225 (d) 113
39. 39.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 625 (b) 875 (c) 635 (d) 1225
40. 40.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x + 1)(x + 45)

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