Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 729 · 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.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9025 (c) 9975 (d) 9035
2. 2.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2525 (b) 3025 (c) 3035 (d) 2025
3. 3.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x + 1)(x + 24)
4. 4.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x − 7)(x − 9)
5. 5.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
6. 6.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 33 (c) 21 (d) 17
7. 7.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 145 (c) 289 (d) 433
8. 8.If a + b = 13 and ab = 22, what is (a − b)²? (a) 125 (b) 9 (c) 169 (d) 81
9. 9.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 14 (c) 9 (d) 8
10. 10.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7744 (c) 7754 (d) 8096
11. 11.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 1089 (b) 909 (c) 729 (d) 1099
12. 12.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 10 (c) 11 (d) 12
13. 13.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 7579 (b) 8091 (c) 7569 (d) 8649
14. 14.What is the product of the roots of x² − 8x + 12 = 0? (a) 13 (b) 12 (c) 11 (d) 8
15. 15.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1859 (b) 1609 (c) 1849 (d) 1369
16. 16.What is the product of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 49 (d) 47
17. 17.What are the roots of x² − 10x + 21 = 0? (a) 1 and 21 (b) 3 and 7 (c) 1 and 9 (d) 2 and 8
18. 18.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 2 and 6 (d) 1 and 7
19. 19.Using (a + b)² = a² + 2ab + b², what is 63²? (a) 3249 (b) 3609 (c) 3969 (d) 3979
20. 20.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 24 (c) 1 and 10 (d) 3 and 8
21. 21.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3844 (c) 3364 (d) 3604
22. 22.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2714 (b) 2304 (c) 2504 (d) 2704
23. 23.What are the roots of x² − 8x + 15 = 0? (a) 1 and 15 (b) 1 and 7 (c) 2 and 6 (d) 3 and 5
24. 24.What are the roots of x² − 7x + 12 = 0? (a) 1 and 6 (b) 1 and 12 (c) 3 and 4 (d) 2 and 6
25. 25.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)
26. 26.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 12 (d) 18
27. 27.If a + b = 7 and ab = 6, what is a² + b²? (a) 61 (b) 43 (c) 49 (d) 37
28. 28.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 137 (c) 7 (d) 49
29. 29.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x − 2)(x − 3) (d) (x + 1)(x + 6)
30. 30.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 91 (c) 1 (d) 121
31. 31.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 208 (c) 352 (d) 160
32. 32.If a + b = 12 and ab = 20, what is a² + b²? (a) 144 (b) 124 (c) 104 (d) 184
33. 33.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 61 (c) 1 (d) 41
34. 34.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
35. 35.If a + b = 5 and ab = 6, what is a² + b²? (a) 19 (b) 25 (c) 37 (d) 13
36. 36.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 313 (c) 137 (d) 181
37. 37.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 1 and 14 (c) 1 and 56 (d) 2 and 28
38. 38.If a + b = 11 and ab = 28, what is a² + b²? (a) 177 (b) 65 (c) 121 (d) 93
39. 39.What is the product of the roots of x² − 9x + 18 = 0? (a) 9 (b) 19 (c) 18 (d) 17
40. 40.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 45) (c) (x − 5)(x − 9) (d) (x + 1)(x + 13)

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