Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 602 · 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 = 14 and ab = 24, what is (a − b)²? (a) 196 (b) 148 (c) 10 (d) 100
2. 2.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 5041 (c) 4761 (d) 4899
3. 3.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1774 (c) 1764 (d) 1604
4. 4.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 19 (d) 13
5. 5.What are the roots of x² − 7x + 12 = 0? (a) 2 and 6 (b) 3 and 4 (c) 1 and 6 (d) 1 and 12
6. 6.What are the roots of x² − 9x + 14 = 0? (a) 1 and 14 (b) 1 and 8 (c) 2 and 7 (d) 3 and 6
7. 7.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3721 (c) 3601 (d) 3731
8. 8.If a + b = 19 and ab = 90, what is a² + b²? (a) 271 (b) 181 (c) 361 (d) 541
9. 9.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 333 (c) 441 (d) 657
10. 10.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 8 (c) 64 (d) 100
11. 11.What is the product of the roots of x² − 8x + 12 = 0? (a) 8 (b) 12 (c) 11 (d) 13
12. 12.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8096 (b) 8464 (c) 7744 (d) 7754
13. 13.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 244 (c) 164 (d) 324
14. 14.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)
15. 15.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 366 (c) 256 (d) 201
16. 16.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4356 (b) 4916 (c) 5486 (d) 5476
17. 17.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)
18. 18.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 15 (c) 8 (d) 7
19. 19.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 9 (c) 121 (d) 3
20. 20.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)
21. 21.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 2 and 24 (c) 1 and 13 (d) 6 and 8
22. 22.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 10 (c) 30 (d) 12
23. 23.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 8 (c) 6 (d) 7
24. 24.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 5 (d) 4
25. 25.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
26. 26.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 45 (c) 13 (d) 14
27. 27.If a + b = 11 and ab = 30, what is (a − b)²? (a) 91 (b) 121 (c) 1 (d) 61
28. 28.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 2 (c) 144 (d) 74
29. 29.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 113 (c) 1 (d) 225
30. 30.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)
31. 31.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1099 (b) 729 (c) 909 (d) 1089
32. 32.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
33. 33.If a + b = 8 and ab = 15, what is a² + b²? (a) 64 (b) 94 (c) 49 (d) 34
34. 34.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 106 (c) 4 (d) 16
35. 35.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 1)(x + 16)
36. 36.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2475 (c) 2035 (d) 2025
37. 37.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
38. 38.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6889 (b) 5929 (c) 6409 (d) 6899
39. 39.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1681 (b) 1601 (c) 1691 (d) 1521
40. 40.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 20 (c) 9 (d) 21

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