Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 786 · 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.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 404 (c) 494 (d) 484
2. 2.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x + 1)(x + 72)
3. 3.If a + b = 13 and ab = 40, what is a² + b²? (a) 89 (b) 249 (c) 129 (d) 169
4. 4.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 1)(x + 10) (c) (x − 4)(x − 7) (d) (x + 4)(x + 7)
5. 5.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 42 (c) 13 (d) 43
6. 6.What is the sum of the roots of x² − 9x + 20 = 0? (a) 10 (b) 8 (c) 20 (d) 9
7. 7.If a + b = 11 and ab = 18, what is a² + b²? (a) 121 (b) 103 (c) 157 (d) 85
8. 8.What is the sum of the roots of x² − 11x + 18 = 0? (a) 18 (b) 11 (c) 10 (d) 12
9. 9.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 5)(x + 8) (d) (x + 1)(x + 12)
10. 10.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 8 (c) 7 (d) 6
11. 11.If a + b = 13 and ab = 42, what is a² + b²? (a) 169 (b) 253 (c) 85 (d) 127
12. 12.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
13. 13.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2025 (c) 3025 (d) 2475
14. 14.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)
15. 15.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 244 (c) 164 (d) 484
16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
17. 17.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
18. 18.If a + b = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 3 (c) 45 (d) 81
19. 19.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
20. 20.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 247 (c) 478 (d) 170
21. 21.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 73 (d) 17
22. 22.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 3 and 21 (d) 7 and 9
23. 23.If a + b = 16 and ab = 55, what is (a − b)²? (a) 6 (b) 256 (c) 146 (d) 36
24. 24.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3731 (c) 3481 (d) 3721
25. 25.What is the product of the roots of x² − 11x + 28 = 0? (a) 27 (b) 11 (c) 28 (d) 29
26. 26.If a + b = 8 and ab = 15, what is a² + b²? (a) 34 (b) 49 (c) 64 (d) 94
27. 27.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1849 (c) 1369 (d) 1609
28. 28.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 1 and 32 (c) 2 and 16 (d) 1 and 11
29. 29.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 13 (c) 12 (d) 36
30. 30.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 12 (c) 14 (d) 13
31. 31.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2116 (b) 2126 (c) 2916 (d) 2484
32. 32.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6084 (b) 6724 (c) 6734 (d) 6404
33. 33.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 169 (c) 125 (d) 213
34. 34.What is the product of the roots of x² − 9x + 14 = 0? (a) 9 (b) 15 (c) 13 (d) 14
35. 35.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 225 (c) 333 (d) 117
36. 36.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9991 (c) 9409 (d) 10609
37. 37.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 3 (c) 89 (d) 9
38. 38.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 3)(x + 8) (c) (x + 1)(x + 24) (d) (x − 3)(x − 8)
39. 39.If a + b = 16 and ab = 48, what is a² + b²? (a) 208 (b) 160 (c) 256 (d) 352
40. 40.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 1 (c) 121 (d) 91

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