Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 39 · 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 + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 2)(x + 4) (c) (x + 1)(x + 5) (d) (x − 2)(x − 4)
2. 2.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2819 (c) 2809 (d) 2209
3. 3.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 289 (c) 223 (d) 157
4. 4.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 971 (b) 841 (c) 961 (d) 901
5. 5.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5184 (c) 4624 (d) 5194
6. 6.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3035 (b) 2025 (c) 3025 (d) 2525
7. 7.If a + b = 11 and ab = 24, what is (a − b)²? (a) 5 (b) 25 (c) 73 (d) 121
8. 8.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6724 (c) 6396 (d) 6094
9. 9.If a + b = 20 and ab = 96, what is (a − b)²? (a) 4 (b) 16 (c) 208 (d) 400
10. 10.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 1156 (c) 916 (d) 1166
11. 11.If a + b = 8 and ab = 15, what is a² + b²? (a) 49 (b) 94 (c) 64 (d) 34
12. 12.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 16 (c) 10 (d) 4
13. 13.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 12 (c) 35 (d) 13
14. 14.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 8 and 9 (c) 1 and 16 (d) 1 and 72
15. 15.What is the product of the roots of x² − 12x + 35 = 0? (a) 35 (b) 36 (c) 12 (d) 34
16. 16.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 193 (c) 529 (d) 277
17. 17.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3604 (c) 3364 (d) 3854
18. 18.If a + b = 9 and ab = 20, what is (a − b)²? (a) 41 (b) 61 (c) 1 (d) 81
19. 19.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 121 (c) 97 (d) 73
20. 20.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x + 1)(x + 6)
21. 21.Which is the correct factorisation of x² + 8x + 12? (a) (x + 1)(x + 7) (b) (x + 1)(x + 12) (c) (x + 2)(x + 6) (d) (x − 2)(x − 6)
22. 22.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 130 (c) 256 (d) 4
23. 23.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
24. 24.What is the product of the roots of x² − 7x + 12 = 0? (a) 13 (b) 12 (c) 7 (d) 11
25. 25.What are the roots of x² − 9x + 20 = 0? (a) 2 and 10 (b) 1 and 20 (c) 4 and 5 (d) 1 and 8
26. 26.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1691 (c) 1521 (d) 1681
27. 27.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 18 (d) 17
28. 28.If a + b = 20 and ab = 96, what is a² + b²? (a) 208 (b) 400 (c) 304 (d) 592
29. 29.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 7 (c) 137 (d) 49
30. 30.What is the product of the roots of x² − 8x + 15 = 0? (a) 14 (b) 8 (c) 15 (d) 16
31. 31.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 11 (c) 9 (d) 10
32. 32.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 4)(x + 8)
33. 33.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9035 (b) 9975 (c) 9025 (d) 11025
34. 34.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 144 (c) 122 (d) 133
35. 35.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)
36. 36.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x − 4)(x − 6) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
37. 37.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
38. 38.What are the roots of x² − 12x + 32 = 0? (a) 1 and 32 (b) 4 and 8 (c) 2 and 16 (d) 1 and 11
39. 39.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)
40. 40.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 170 (d) 324

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