Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 440 · 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 + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
2. 2.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 1)(x + 36)
3. 3.What is the product of the roots of x² − 12x + 27 = 0? (a) 12 (b) 27 (c) 28 (d) 26
4. 4.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x − 5)(x − 7) (c) (x + 5)(x + 7) (d) (x + 1)(x + 35)
5. 5.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 25 (c) 5 (d) 193
6. 6.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 484 (d) 244
7. 7.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x − 2)(x − 5) (c) (x + 1)(x + 10) (d) (x + 1)(x + 6)
8. 8.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 7754 (b) 8096 (c) 7744 (d) 8464
9. 9.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2809 (c) 2509 (d) 2209
10. 10.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 256 (c) 416 (d) 576
11. 11.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 2 and 10 (c) 5 and 7 (d) 1 and 11
12. 12.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x + 2)(x + 4) (c) (x − 2)(x − 4) (d) (x + 1)(x + 8)
13. 13.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 7406 (c) 8836 (d) 7396
14. 14.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 1 and 14 (c) 7 and 8 (d) 2 and 28
15. 15.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4896 (c) 4624 (d) 4634
16. 16.If a + b = 15 and ab = 54, what is (a − b)²? (a) 3 (b) 9 (c) 225 (d) 117
17. 17.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 18 (d) 17
18. 18.What are the roots of x² − 10x + 16 = 0? (a) 3 and 7 (b) 1 and 9 (c) 1 and 16 (d) 2 and 8
19. 19.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 686 (b) 1156 (c) 676 (d) 884
20. 20.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1296 (d) 1306
21. 21.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 54 (c) 1 and 14 (d) 6 and 9
22. 22.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
23. 23.If a + b = 14 and ab = 45, what is (a − b)²? (a) 106 (b) 196 (c) 16 (d) 4
24. 24.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
25. 25.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 6)(x + 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
26. 26.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 851 (c) 899 (d) 841
27. 27.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 4 (c) 202 (d) 2
28. 28.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x − 2)(x − 7) (c) (x + 1)(x + 14) (d) (x + 2)(x + 7)
29. 29.What is the sum of the roots of x² − 8x + 15 = 0? (a) 9 (b) 7 (c) 8 (d) 15
30. 30.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 6 and 8 (c) 1 and 48 (d) 2 and 24
31. 31.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 137 (c) 225 (d) 49
32. 32.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 69 (c) 49 (d) 29
33. 33.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
34. 34.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2484 (c) 2116 (d) 2126
35. 35.If a + b = 5 and ab = 6, what is a² + b²? (a) 37 (b) 25 (c) 19 (d) 13
36. 36.What is the sum of the roots of x² − 8x + 12 = 0? (a) 7 (b) 12 (c) 9 (d) 8
37. 37.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
38. 38.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4899 (b) 5041 (c) 4761 (d) 4771
39. 39.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1849 (c) 1609 (d) 1859
40. 40.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 13 (c) 15 (d) 48

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