Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 127 · 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 + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
2. 2.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 6 and 8 (c) 1 and 48 (d) 2 and 24
3. 3.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 73 (c) 17 (d) 72
4. 4.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2025 (c) 2035 (d) 2475
5. 5.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3035 (b) 3575 (c) 3025 (d) 4225
6. 6.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 26 (c) 28 (d) 12
7. 7.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 2)(x + 8)
8. 8.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1444 (b) 1596 (c) 1454 (d) 1764
9. 9.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
10. 10.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 9 (d) 10
11. 11.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 12 (c) 32 (d) 33
12. 12.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 5625 (b) 7225 (c) 6375 (d) 5635
13. 13.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1444 (b) 1774 (c) 1604 (d) 1764
14. 14.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 17 (d) 16
15. 15.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x + 1)(x + 21) (c) (x − 3)(x − 7) (d) (x + 3)(x + 7)
16. 16.If a + b = 16 and ab = 55, what is a² + b²? (a) 256 (b) 201 (c) 146 (d) 366
17. 17.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)
18. 18.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 4489 (b) 5339 (c) 5329 (d) 4909
19. 19.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1166 (c) 916 (d) 1156
20. 20.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 34 (c) 35 (d) 12
21. 21.If a + b = 13 and ab = 36, what is a² + b²? (a) 133 (b) 241 (c) 97 (d) 169
22. 22.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 100 (c) 52 (d) 4
23. 23.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 9 (d) 225
24. 24.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 9409 (c) 10609 (d) 10619
25. 25.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 11 (c) 18 (d) 10
26. 26.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 47 (c) 14 (d) 48
27. 27.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 20 (c) 19 (d) 9
28. 28.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 23 (c) 24 (d) 11
29. 29.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 529 (c) 193 (d) 361
30. 30.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 28 (c) 12 (d) 11
31. 31.What is the product of the roots of x² − 10x + 24 = 0? (a) 25 (b) 24 (c) 10 (d) 23
32. 32.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x + 1)(x + 48) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
33. 33.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 2 and 36 (d) 8 and 9
34. 34.What is the sum of the roots of x² − 13x + 40 = 0? (a) 13 (b) 40 (c) 14 (d) 12
35. 35.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2504 (c) 2304 (d) 2704
36. 36.If a + b = 8 and ab = 7, what is a² + b²? (a) 50 (b) 78 (c) 64 (d) 57
37. 37.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 1 and 18 (c) 1 and 8 (d) 2 and 9
38. 38.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
39. 39.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 9 (c) 7 (d) 1
40. 40.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 12 (c) 24 (d) 11

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