Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 613 · 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 are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 8 and 9 (c) 2 and 36 (d) 1 and 72
2. 2.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 784 (b) 904 (c) 1034 (d) 1024
3. 3.If a + b = 4 and ab = 3, what is (a − b)²? (a) 2 (b) 10 (c) 4 (d) 16
4. 4.If a + b = 12 and ab = 32, what is (a − b)²? (a) 80 (b) 144 (c) 4 (d) 16
5. 5.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3136 (b) 4106 (c) 3616 (d) 4096
6. 6.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)
7. 7.If a + b = 7 and ab = 12, what is a² + b²? (a) 73 (b) 25 (c) 49 (d) 37
8. 8.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1575 (d) 1235
9. 9.What is the sum of the roots of x² − 7x + 12 = 0? (a) 8 (b) 7 (c) 6 (d) 12
10. 10.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 971 (c) 841 (d) 961
11. 11.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x − 3)(x − 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
12. 12.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 10) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
13. 13.If a + b = 11 and ab = 10, what is a² + b²? (a) 101 (b) 111 (c) 121 (d) 141
14. 14.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 9 (c) 20 (d) 19
15. 15.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 2 and 21 (c) 6 and 7 (d) 1 and 12
16. 16.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 598 (c) 400 (d) 202
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.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 42 (c) 43 (d) 41
19. 19.If a + b = 11 and ab = 18, what is (a − b)²? (a) 49 (b) 7 (c) 121 (d) 85
20. 20.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1369 (c) 1849 (d) 1859
21. 21.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 271 (c) 181 (d) 361
22. 22.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 8 (c) 1 and 18 (d) 3 and 6
23. 23.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 841 (c) 899 (d) 961
24. 24.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 729 (c) 739 (d) 891
25. 25.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 401 (c) 361 (d) 451
26. 26.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
27. 27.What is the sum of the roots of x² − 10x + 21 = 0? (a) 21 (b) 9 (c) 11 (d) 10
28. 28.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 15 (d) 16
29. 29.What is the product of the roots of x² − 6x + 8 = 0? (a) 9 (b) 6 (c) 8 (d) 7
30. 30.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 6384 (b) 5786 (c) 7056 (d) 5776
31. 31.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 10) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
32. 32.What are the roots of x² − 13x + 40 = 0? (a) 5 and 8 (b) 1 and 40 (c) 1 and 12 (d) 2 and 20
33. 33.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 4 (d) 16
34. 34.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
35. 35.If a + b = 18 and ab = 72, what is (a − b)²? (a) 36 (b) 6 (c) 180 (d) 324
36. 36.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 11) (b) (x + 4)(x + 8) (c) (x − 4)(x − 8) (d) (x + 1)(x + 32)
37. 37.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)
38. 38.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4356 (b) 4366 (c) 5476 (d) 4884
39. 39.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3491 (b) 3599 (c) 3481 (d) 3721
40. 40.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2209 (b) 2809 (c) 2219 (d) 2491

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