Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 48 · 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 sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 14 (d) 16
2. 2.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 7235 (c) 8075 (d) 9025
3. 3.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1849 (d) 1609
4. 4.If a + b = 9 and ab = 20, what is a² + b²? (a) 121 (b) 41 (c) 81 (d) 61
5. 5.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 196 (c) 100 (d) 2
6. 6.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)
7. 7.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 5 and 7 (c) 2 and 10 (d) 1 and 11
8. 8.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 6 (c) 2 and 6 (d) 1 and 12
9. 9.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1166 (c) 1156 (d) 676
10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
11. 11.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 904 (c) 1034 (d) 1024
12. 12.What are the roots of x² − 17x + 72 = 0? (a) 2 and 36 (b) 1 and 72 (c) 1 and 16 (d) 8 and 9
13. 13.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 598 (d) 400
14. 14.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2601 (b) 2401 (c) 2501 (d) 2611
15. 15.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 35 (c) 12 (d) 34
16. 16.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
17. 17.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6084 (c) 6724 (d) 6404
18. 18.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 7569 (c) 8109 (d) 8649
19. 19.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 36 (c) 12 (d) 13
20. 20.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 81 (c) 1 (d) 41
21. 21.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 13 (c) 43 (d) 42
22. 22.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 433 (c) 289 (d) 217
23. 23.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 676 (c) 686 (d) 1156
24. 24.What is the sum of the roots of x² − 14x + 45 = 0? (a) 45 (b) 15 (c) 14 (d) 13
25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
26. 26.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
27. 27.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1089 (c) 1099 (d) 729
28. 28.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 8 (d) 5
29. 29.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 18) (c) (x − 2)(x − 9) (d) (x + 1)(x + 10)
30. 30.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1454 (b) 1444 (c) 1764 (d) 1596
31. 31.If a + b = 11 and ab = 28, what is a² + b²? (a) 65 (b) 93 (c) 121 (d) 177
32. 32.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x − 6)(x − 7) (c) (x + 6)(x + 7) (d) (x + 1)(x + 12)
33. 33.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 6 (c) 36 (d) 256
34. 34.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 11 (c) 12 (d) 27
35. 35.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x − 3)(x − 4) (c) (x + 1)(x + 12) (d) (x + 3)(x + 4)
36. 36.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 127 (c) 85 (d) 169
37. 37.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 37 (c) 49 (d) 73
38. 38.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
39. 39.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)
40. 40.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 2 (c) 324 (d) 4

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