Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 899 · 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.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 13 (c) 15 (d) 14
2. 2.If a + b = 15 and ab = 56, what is (a − b)²? (a) 225 (b) 169 (c) 1 (d) 113
3. 3.Using (a + b)² = a² + 2ab + b², what is 32²? (a) 904 (b) 784 (c) 1024 (d) 1034
4. 4.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 10826 (c) 9216 (d) 10016
5. 5.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 10201 (c) 9801 (d) 9999
6. 6.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 537 (c) 361 (d) 185
7. 7.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
8. 8.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2116 (c) 2916 (d) 2484
9. 9.If a + b = 3 and ab = 2, what is a² + b²? (a) 9 (b) 13 (c) 5 (d) 7
10. 10.What are the roots of x² − 11x + 30 = 0? (a) 5 and 6 (b) 1 and 30 (c) 1 and 10 (d) 2 and 15
11. 11.What is the sum of the roots of x² − 15x + 56 = 0? (a) 16 (b) 56 (c) 14 (d) 15
12. 12.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 1 (c) 61 (d) 91
13. 13.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 18) (c) (x + 1)(x + 8) (d) (x + 3)(x + 6)
14. 14.If a + b = 8 and ab = 15, what is a² + b²? (a) 94 (b) 49 (c) 34 (d) 64
15. 15.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)
16. 16.What is the product of the roots of x² − 11x + 28 = 0? (a) 28 (b) 27 (c) 29 (d) 11
17. 17.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 196 (c) 8 (d) 130
18. 18.If a + b = 17 and ab = 66, what is a² + b²? (a) 223 (b) 421 (c) 157 (d) 289
19. 19.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 5486 (c) 4356 (d) 5476
20. 20.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 961 (b) 841 (c) 971 (d) 901
21. 21.Which is the correct factorisation of x² + 10x + 16? (a) (x + 2)(x + 8) (b) (x − 2)(x − 8) (c) (x + 1)(x + 9) (d) (x + 1)(x + 16)
22. 22.What is the sum of the roots of x² − 11x + 30 = 0? (a) 12 (b) 30 (c) 10 (d) 11
23. 23.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 2 (c) 52 (d) 4
24. 24.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 5184 (b) 4896 (c) 4634 (d) 4624
25. 25.If a + b = 20 and ab = 99, what is a² + b²? (a) 598 (b) 202 (c) 301 (d) 400
26. 26.Which is the correct factorisation of x² + 7x + 12? (a) (x − 3)(x − 4) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x + 1)(x + 12)
27. 27.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2035 (c) 2025 (d) 3025
28. 28.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
29. 29.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 63 (c) 7 and 9 (d) 1 and 15
30. 30.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 40 (c) 39 (d) 13
31. 31.What are the roots of x² − 14x + 45 = 0? (a) 1 and 13 (b) 5 and 9 (c) 3 and 15 (d) 1 and 45
32. 32.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 1225 (b) 625 (c) 635 (d) 875
33. 33.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 494 (b) 324 (c) 484 (d) 404
34. 34.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 2 and 21 (d) 1 and 12
35. 35.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 47 (c) 49 (d) 48
36. 36.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 149 (c) 9 (d) 289
37. 37.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 13 (c) 12 (d) 27
38. 38.If a + b = 14 and ab = 45, what is a² + b²? (a) 106 (b) 286 (c) 196 (d) 151
39. 39.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 4)(x + 6) (d) (x + 1)(x + 9)
40. 40.If a + b = 15 and ab = 44, what is a² + b²? (a) 137 (b) 313 (c) 225 (d) 181

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