Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 496 · 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 + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 1)(x + 56)
2. 2.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)
3. 3.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
4. 4.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2809 (c) 2509 (d) 2209
5. 5.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 4 (d) 2
6. 6.What is the sum of the roots of x² − 11x + 28 = 0? (a) 28 (b) 10 (c) 11 (d) 12
7. 7.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 3)(x + 8) (d) (x + 1)(x + 24)
8. 8.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 16 (c) 14 (d) 15
9. 9.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 10 (c) 24 (d) 25
10. 10.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 13 (c) 27 (d) 12
11. 11.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)
12. 12.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 18 (c) 19 (d) 11
13. 13.What is the sum of the roots of x² − 9x + 14 = 0? (a) 10 (b) 8 (c) 9 (d) 14
14. 14.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 17 (d) 73
15. 15.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10619 (c) 9409 (d) 10609
16. 16.What are the roots of x² − 9x + 18 = 0? (a) 2 and 9 (b) 1 and 18 (c) 1 and 8 (d) 3 and 6
17. 17.If a + b = 13 and ab = 36, what is a² + b²? (a) 169 (b) 241 (c) 133 (d) 97
18. 18.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 3 (c) 25 (d) 17
19. 19.If a + b = 16 and ab = 60, what is (a − b)²? (a) 16 (b) 4 (c) 256 (d) 136
20. 20.What is the sum of the roots of x² − 10x + 16 = 0? (a) 10 (b) 9 (c) 16 (d) 11
21. 21.Which is the correct factorisation of x² + 9x + 20? (a) (x + 4)(x + 5) (b) (x + 1)(x + 8) (c) (x + 1)(x + 20) (d) (x − 4)(x − 5)
22. 22.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 27 (c) 1 and 11 (d) 3 and 9
23. 23.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 65 (c) 93 (d) 177
24. 24.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 13 (c) 12 (d) 36
25. 25.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4884 (c) 4356 (d) 4366
26. 26.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 1 (c) 13 (d) 19
27. 27.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 3 (c) 9 (d) 49
28. 28.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
29. 29.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 7569 (c) 8659 (d) 8109
30. 30.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 5776 (c) 7056 (d) 6416
31. 31.If a + b = 17 and ab = 66, what is a² + b²? (a) 157 (b) 223 (c) 289 (d) 421
32. 32.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 3035 (c) 2525 (d) 3025
33. 33.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 676 (b) 916 (c) 1156 (d) 1166
34. 34.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 12)
35. 35.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)
36. 36.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 12 (c) 10 (d) 11
37. 37.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 54 (c) 53 (d) 55
38. 38.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9216 (c) 9226 (d) 9984
39. 39.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
40. 40.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 12 (c) 11 (d) 32

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