Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 283 · 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 − 17x + 72 = 0? (a) 17 (b) 72 (c) 71 (d) 73
2. 2.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
3. 3.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 41 (c) 61 (d) 121
4. 4.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 1)(x + 11) (c) (x + 3)(x + 9) (d) (x − 3)(x − 9)
5. 5.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 202 (d) 4
6. 6.If a + b = 16 and ab = 60, what is a² + b²? (a) 256 (b) 376 (c) 196 (d) 136
7. 7.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 20 (c) 8 (d) 9
8. 8.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
9. 9.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 145 (d) 217
10. 10.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
11. 11.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 130 (c) 256 (d) 4
12. 12.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4366 (b) 4356 (c) 4884 (d) 5476
13. 13.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5041 (b) 5051 (c) 4901 (d) 4761
14. 14.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3575 (b) 4225 (c) 3025 (d) 3035
15. 15.If a + b = 12 and ab = 32, what is (a − b)²? (a) 16 (b) 4 (c) 144 (d) 80
16. 16.If a + b = 3 and ab = 2, what is (a − b)²? (a) 9 (b) 7 (c) 1 (d) 5
17. 17.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10826 (c) 10816 (d) 10016
18. 18.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 16 (c) 54 (d) 15
19. 19.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)
20. 20.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 2 (c) 36 (d) 4
21. 21.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 72 (d) 18
22. 22.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 63 (d) 17
23. 23.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
24. 24.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 3 (c) 17 (d) 25
25. 25.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
26. 26.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6084 (c) 6724 (d) 6734
27. 27.If a + b = 14 and ab = 33, what is (a − b)²? (a) 196 (b) 130 (c) 8 (d) 64
28. 28.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10619 (b) 10609 (c) 10009 (d) 9409
29. 29.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x − 8)(x − 9)
30. 30.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 2475 (d) 3025
31. 31.What is the product of the roots of x² − 11x + 30 = 0? (a) 29 (b) 11 (c) 31 (d) 30
32. 32.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
33. 33.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6889 (c) 5929 (d) 6391
34. 34.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 4761 (c) 5041 (d) 4771
35. 35.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4891 (c) 4499 (d) 4489
36. 36.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
37. 37.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
38. 38.If a + b = 9 and ab = 14, what is a² + b²? (a) 53 (b) 109 (c) 67 (d) 81
39. 39.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 6)(x + 9) (c) (x + 1)(x + 14) (d) (x − 6)(x − 9)
40. 40.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 400 (c) 301 (d) 202

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