Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 112 · 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.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
2. 2.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 851 (c) 899 (d) 841
3. 3.What is the product of the roots of x² − 12x + 27 = 0? (a) 28 (b) 12 (c) 27 (d) 26
4. 4.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 1)(x + 8) (c) (x − 4)(x − 5) (d) (x + 4)(x + 5)
5. 5.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 40 (c) 16 (d) 4
6. 6.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 6 and 8 (c) 1 and 13 (d) 2 and 24
7. 7.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 277 (d) 193
8. 8.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
9. 9.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 225 (c) 441 (d) 3
10. 10.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2401 (c) 2611 (d) 2601
11. 11.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 4096 (c) 3136 (d) 3616
12. 12.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1225 (b) 1625 (c) 2025 (d) 2035
13. 13.If a + b = 13 and ab = 42, what is (a − b)²? (a) 127 (b) 85 (c) 169 (d) 1
14. 14.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1774 (c) 1764 (d) 1444
15. 15.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 4)(x + 7) (c) (x − 4)(x − 7) (d) (x + 1)(x + 28)
16. 16.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 18 (d) 12
17. 17.If a + b = 9 and ab = 14, what is (a − b)²? (a) 53 (b) 25 (c) 81 (d) 5
18. 18.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 30 (c) 1 and 10 (d) 5 and 6
19. 19.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1521 (c) 1531 (d) 1681
20. 20.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 181 (c) 1 (d) 361
21. 21.If a + b = 12 and ab = 32, what is (a − b)²? (a) 144 (b) 16 (c) 80 (d) 4
22. 22.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 441 (c) 361 (d) 401
23. 23.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 11 (d) 12
24. 24.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 130 (c) 2 (d) 4
25. 25.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 15 (d) 56
26. 26.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 4891 (b) 4489 (c) 4499 (d) 5329
27. 27.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 2)(x + 4) (c) (x + 1)(x + 8) (d) (x + 1)(x + 5)
28. 28.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 160 (c) 256 (d) 8
29. 29.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 334 (c) 484 (d) 396
30. 30.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 6 (c) 256 (d) 36
31. 31.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 157 (c) 289 (d) 421
32. 32.If a + b = 18 and ab = 77, what is a² + b²? (a) 478 (b) 247 (c) 170 (d) 324
33. 33.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 5051 (c) 4761 (d) 4901
34. 34.What is the sum of the roots of x² − 6x + 8 = 0? (a) 7 (b) 5 (c) 8 (d) 6
35. 35.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 193 (c) 361 (d) 5
36. 36.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 494 (c) 324 (d) 404
37. 37.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)
38. 38.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 16 (c) 62 (d) 63
39. 39.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 127 (d) 85
40. 40.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x − 7)(x − 9) (c) (x + 1)(x + 15) (d) (x + 1)(x + 63)

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