Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 823 · 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.If a + b = 5 and ab = 6, what is a² + b²? (a) 25 (b) 13 (c) 19 (d) 37
2. 2.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 4)(x + 5) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
3. 3.What are the roots of x² − 13x + 42 = 0? (a) 1 and 12 (b) 2 and 21 (c) 6 and 7 (d) 1 and 42
4. 4.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1599 (c) 1531 (d) 1521
5. 5.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 256 (c) 416 (d) 576
6. 6.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5939 (c) 5929 (d) 6889
7. 7.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6094 (c) 6724 (d) 6084
8. 8.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
9. 9.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 65 (d) 177
10. 10.If a + b = 16 and ab = 48, what is (a − b)²? (a) 256 (b) 8 (c) 64 (d) 160
11. 11.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
12. 12.If a + b = 10 and ab = 9, what is (a − b)²? (a) 100 (b) 64 (c) 82 (d) 8
13. 13.What is the sum of the roots of x² − 9x + 14 = 0? (a) 9 (b) 8 (c) 10 (d) 14
14. 14.If a + b = 6 and ab = 8, what is (a − b)²? (a) 2 (b) 36 (c) 20 (d) 4
15. 15.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 71 (d) 17
16. 16.If a + b = 9 and ab = 14, what is a² + b²? (a) 81 (b) 109 (c) 67 (d) 53
17. 17.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 2 and 28 (d) 1 and 56
18. 18.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 48) (c) (x − 6)(x − 8) (d) (x + 1)(x + 13)
19. 19.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1936 (c) 1616 (d) 1946
20. 20.If a + b = 12 and ab = 11, what is a² + b²? (a) 133 (b) 144 (c) 166 (d) 122
21. 21.If a + b = 17 and ab = 66, what is (a − b)²? (a) 289 (b) 5 (c) 25 (d) 157
22. 22.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 3 and 6 (c) 1 and 8 (d) 1 and 18
23. 23.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
24. 24.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
25. 25.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2509 (b) 2819 (c) 2209 (d) 2809
26. 26.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
27. 27.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
28. 28.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 9604 (b) 10004 (c) 10404 (d) 10414
29. 29.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 5776 (b) 7066 (c) 7056 (d) 6416
30. 30.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 11 (c) 19 (d) 18
31. 31.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 361 (c) 271 (d) 1
32. 32.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 292 (c) 100 (d) 148
33. 33.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 11025 (c) 10025 (d) 9025
34. 34.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6404 (b) 6734 (c) 6084 (d) 6724
35. 35.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 16 (c) 18 (d) 72
36. 36.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 1)(x + 10) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
37. 37.What is the sum of the roots of x² − 10x + 24 = 0? (a) 10 (b) 11 (c) 9 (d) 24
38. 38.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 9025 (c) 9035 (d) 7225
39. 39.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 14 (c) 16 (d) 56
40. 40.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 1)(x + 16)

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