Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 785 · 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.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 40 (c) 13 (d) 39
2. 2.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 4624 (c) 5194 (d) 5184
3. 3.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 7931 (b) 8281 (c) 7921 (d) 8099
4. 4.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 289 (c) 229 (d) 169
5. 5.If a + b = 14 and ab = 33, what is (a − b)²? (a) 8 (b) 196 (c) 64 (d) 130
6. 6.If a + b = 18 and ab = 77, what is (a − b)²? (a) 16 (b) 4 (c) 324 (d) 170
7. 7.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
8. 8.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 2 and 3 (c) 1 and 4 (d) 1 and 6
9. 9.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 52 (c) 76 (d) 100
10. 10.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 6)(x + 9) (d) (x + 1)(x + 14)
11. 11.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 3 (c) 25 (d) 9
12. 12.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x − 5)(x − 9) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
13. 13.If a + b = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 57 (d) 50
14. 14.If a + b = 16 and ab = 60, what is a² + b²? (a) 196 (b) 256 (c) 136 (d) 376
15. 15.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 100 (c) 4 (d) 2
16. 16.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x − 3)(x − 6)
17. 17.If a + b = 9 and ab = 18, what is (a − b)²? (a) 3 (b) 81 (c) 9 (d) 45
18. 18.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 20 (c) 1 and 8 (d) 4 and 5
19. 19.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 145 (d) 289
20. 20.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4624 (c) 4896 (d) 4634
21. 21.If a + b = 18 and ab = 80, what is a² + b²? (a) 164 (b) 484 (c) 244 (d) 324
22. 22.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 592 (c) 304 (d) 400
23. 23.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2116 (c) 2926 (d) 2516
24. 24.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 71 (d) 17
25. 25.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 5929 (c) 6889 (d) 6899
26. 26.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 8 (c) 9 (d) 14
27. 27.If a + b = 9 and ab = 18, what is a² + b²? (a) 63 (b) 81 (c) 45 (d) 117
28. 28.What is the sum of the roots of x² − 12x + 32 = 0? (a) 32 (b) 13 (c) 11 (d) 12
29. 29.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1521 (c) 1531 (d) 1599
30. 30.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
31. 31.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2116 (c) 2126 (d) 2484
32. 32.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 539 (c) 409 (d) 529
33. 33.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)
34. 34.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
35. 35.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
36. 36.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 4225 (b) 3035 (c) 3575 (d) 3025
37. 37.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 7225 (c) 9025 (d) 9035
38. 38.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 2 and 16 (c) 1 and 11 (d) 1 and 32
39. 39.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 8 and 9 (c) 1 and 16 (d) 2 and 36
40. 40.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)

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