Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 791 · 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 23²? (a) 409 (b) 529 (c) 289 (d) 539
2. 2.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 16 (c) 9 (d) 10
3. 3.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8281 (c) 8291 (d) 8101
4. 4.What is the product of the roots of x² − 13x + 36 = 0? (a) 13 (b) 35 (c) 36 (d) 37
5. 5.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 841 (d) 851
6. 6.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x + 4)(x + 6) (c) (x − 4)(x − 6) (d) (x + 1)(x + 9)
7. 7.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6724 (c) 6404 (d) 6084
8. 8.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 324 (b) 494 (c) 484 (d) 404
9. 9.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
10. 10.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
11. 11.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 57 (d) 56
12. 12.What is the product of the roots of x² − 10x + 21 = 0? (a) 10 (b) 21 (c) 20 (d) 22
13. 13.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 256 (c) 130 (d) 382
14. 14.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10001 (b) 9801 (c) 10201 (d) 10211
15. 15.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9999 (c) 9801 (d) 9811
16. 16.What is the product of the roots of x² − 17x + 72 = 0? (a) 72 (b) 73 (c) 17 (d) 71
17. 17.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 7 and 8 (c) 2 and 28 (d) 1 and 14
18. 18.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
19. 19.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 55 (d) 54
20. 20.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
21. 21.What is the product of the roots of x² − 11x + 24 = 0? (a) 11 (b) 25 (c) 23 (d) 24
22. 22.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 225 (c) 635 (d) 625
23. 23.If a + b = 11 and ab = 30, what is (a − b)²? (a) 61 (b) 121 (c) 1 (d) 91
24. 24.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 289 (c) 3 (d) 9
25. 25.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x − 3)(x − 5) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
26. 26.If a + b = 12 and ab = 35, what is (a − b)²? (a) 4 (b) 144 (c) 74 (d) 2
27. 27.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 13) (b) (x − 6)(x − 8) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
28. 28.If a + b = 10 and ab = 9, what is a² + b²? (a) 82 (b) 100 (c) 91 (d) 118
29. 29.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
30. 30.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1089 (c) 1099 (d) 729
31. 31.What is the product of the roots of x² − 13x + 40 = 0? (a) 40 (b) 39 (c) 13 (d) 41
32. 32.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 2 and 27 (c) 1 and 14 (d) 6 and 9
33. 33.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 16 (c) 324 (d) 170
34. 34.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 6 (c) 8 (d) 7
35. 35.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1454 (d) 1444
36. 36.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
37. 37.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)
38. 38.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 1 and 11 (c) 2 and 10 (d) 3 and 9
39. 39.If a + b = 16 and ab = 55, what is a² + b²? (a) 201 (b) 256 (c) 366 (d) 146
40. 40.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 324 (c) 334 (d) 484

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