Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 330 · 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 + 15x + 54? (a) (x + 1)(x + 14) (b) (x − 6)(x − 9) (c) (x + 1)(x + 54) (d) (x + 6)(x + 9)
2. 2.If a + b = 13 and ab = 30, what is (a − b)²? (a) 7 (b) 169 (c) 49 (d) 109
3. 3.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 5)(x + 9) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
4. 4.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 36 (c) 100 (d) 68
5. 5.What is the sum of the roots of x² − 13x + 42 = 0? (a) 12 (b) 42 (c) 14 (d) 13
6. 6.What is the sum of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 5 (d) 7
7. 7.If a + b = 11 and ab = 28, what is (a − b)²? (a) 65 (b) 121 (c) 9 (d) 3
8. 8.What are the roots of x² − 15x + 56 = 0? (a) 1 and 14 (b) 1 and 56 (c) 7 and 8 (d) 2 and 28
9. 9.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 9) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x − 4)(x − 6)
10. 10.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x + 4)(x + 5) (c) (x − 4)(x − 5) (d) (x + 1)(x + 8)
11. 11.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 9 (c) 7 (d) 10
12. 12.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
13. 13.What are the roots of x² − 12x + 35 = 0? (a) 1 and 35 (b) 2 and 10 (c) 1 and 11 (d) 5 and 7
14. 14.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1616 (c) 1296 (d) 1936
15. 15.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1521 (c) 1681 (d) 1691
16. 16.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 899 (c) 841 (d) 961
17. 17.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 4)(x + 8) (c) (x + 1)(x + 11) (d) (x + 1)(x + 32)
18. 18.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3364 (c) 3854 (d) 3604
19. 19.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 3 (c) 185 (d) 9
20. 20.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 35 (c) 13 (d) 12
21. 21.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 478 (c) 170 (d) 324
22. 22.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 69 (c) 39 (d) 29
23. 23.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x + 1)(x + 16) (c) (x − 2)(x − 8) (d) (x + 1)(x + 9)
24. 24.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 4 (c) 36 (d) 20
25. 25.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 28 (c) 12 (d) 26
26. 26.If a + b = 10 and ab = 24, what is (a − b)²? (a) 4 (b) 2 (c) 100 (d) 52
27. 27.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 6416 (c) 7066 (d) 7056
28. 28.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)
29. 29.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 55 (c) 15 (d) 57
30. 30.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 249 (c) 89 (d) 129
31. 31.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 12) (b) (x + 4)(x + 9) (c) (x + 1)(x + 36) (d) (x − 4)(x − 9)
32. 32.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x − 7)(x − 9) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
33. 33.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6889 (c) 6409 (d) 5929
34. 34.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4771 (b) 4761 (c) 4899 (d) 5041
35. 35.If a + b = 9 and ab = 20, what is a² + b²? (a) 81 (b) 61 (c) 121 (d) 41
36. 36.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 169 (c) 9 (d) 81
37. 37.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 16 (d) 18
38. 38.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 7225 (c) 9035 (d) 9025
39. 39.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 15) (d) (x + 1)(x + 7)
40. 40.Which is the correct factorisation of x² + 13x + 40? (a) (x + 5)(x + 8) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 1)(x + 40)

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