Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 485 · 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 = 9 and ab = 18, what is (a − b)²? (a) 9 (b) 45 (c) 3 (d) 81
2. 2.What is the product of the roots of x² − 13x + 42 = 0? (a) 42 (b) 43 (c) 41 (d) 13
3. 3.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 5) (b) (x − 2)(x − 4) (c) (x + 1)(x + 8) (d) (x + 2)(x + 4)
4. 4.If a + b = 15 and ab = 44, what is a² + b²? (a) 137 (b) 313 (c) 181 (d) 225
5. 5.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 15) (c) (x − 7)(x − 9) (d) (x + 1)(x + 63)
6. 6.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 3)(x + 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
7. 7.If a + b = 21 and ab = 108, what is (a − b)²? (a) 9 (b) 3 (c) 441 (d) 225
8. 8.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 1 and 10 (c) 1 and 24 (d) 3 and 8
9. 9.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 181 (c) 361 (d) 271
10. 10.What are the roots of x² − 16x + 63 = 0? (a) 1 and 63 (b) 1 and 15 (c) 7 and 9 (d) 3 and 21
11. 11.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 13) (d) (x + 1)(x + 48)
12. 12.What is the product of the roots of x² − 10x + 24 = 0? (a) 23 (b) 25 (c) 24 (d) 10
13. 13.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 409 (b) 289 (c) 539 (d) 529
14. 14.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 1)(x + 12) (c) (x + 1)(x + 36) (d) (x + 4)(x + 9)
15. 15.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 7 (c) 5 (d) 6
16. 16.Using (a + b)² = a² + 2ab + b², what is 73²? (a) 5339 (b) 4909 (c) 4489 (d) 5329
17. 17.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
18. 18.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x + 1)(x + 28)
19. 19.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 1)(x + 8) (d) (x + 4)(x + 5)
20. 20.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x + 1)(x + 27) (c) (x − 3)(x − 9) (d) (x + 3)(x + 9)
21. 21.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
22. 22.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1089 (c) 1099 (d) 909
23. 23.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 57 (d) 56
24. 24.If a + b = 11 and ab = 30, what is a² + b²? (a) 121 (b) 181 (c) 61 (d) 91
25. 25.If a + b = 17 and ab = 66, what is (a − b)²? (a) 5 (b) 157 (c) 289 (d) 25
26. 26.What is the sum of the roots of x² − 9x + 18 = 0? (a) 18 (b) 10 (c) 8 (d) 9
27. 27.What is the sum of the roots of x² − 11x + 28 = 0? (a) 11 (b) 12 (c) 10 (d) 28
28. 28.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 739 (b) 729 (c) 891 (d) 1089
29. 29.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 7225 (c) 9035 (d) 9025
30. 30.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 54 (c) 15 (d) 53
31. 31.What are the roots of x² − 11x + 18 = 0? (a) 2 and 9 (b) 1 and 10 (c) 3 and 6 (d) 1 and 18
32. 32.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10826 (c) 10016 (d) 10816
33. 33.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 13 (d) 12
34. 34.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6094 (c) 6084 (d) 6724
35. 35.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)
36. 36.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 875 (c) 635 (d) 625
37. 37.If a + b = 16 and ab = 48, what is a² + b²? (a) 256 (b) 352 (c) 160 (d) 208
38. 38.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 1 and 14 (c) 6 and 9 (d) 1 and 54
39. 39.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 12 (c) 10 (d) 24
40. 40.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6241 (c) 6561 (d) 6401

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