Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 280 · 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 82²? (a) 6724 (b) 6734 (c) 6084 (d) 6404
2. 2.If a + b = 9 and ab = 8, what is a² + b²? (a) 81 (b) 65 (c) 73 (d) 97
3. 3.What is the sum of the roots of x² − 12x + 32 = 0? (a) 13 (b) 12 (c) 32 (d) 11
4. 4.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 1024 (d) 896
5. 5.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
6. 6.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
7. 7.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 8075 (c) 7225 (d) 9025
8. 8.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 256 (c) 196 (d) 136
9. 9.If a + b = 16 and ab = 63, what is (a − b)²? (a) 4 (b) 2 (c) 256 (d) 130
10. 10.What is the sum of the roots of x² − 5x + 6 = 0? (a) 4 (b) 7 (c) 5 (d) 6
11. 11.If a + b = 10 and ab = 24, what is a² + b²? (a) 148 (b) 76 (c) 52 (d) 100
12. 12.What are the roots of x² − 11x + 18 = 0? (a) 1 and 18 (b) 1 and 10 (c) 3 and 6 (d) 2 and 9
13. 13.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 1)(x + 13)
14. 14.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 42) (d) (x + 1)(x + 12)
15. 15.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8281 (b) 7921 (c) 8101 (d) 8291
16. 16.What are the roots of x² − 14x + 45 = 0? (a) 3 and 15 (b) 1 and 13 (c) 1 and 45 (d) 5 and 9
17. 17.What are the roots of x² − 10x + 21 = 0? (a) 2 and 8 (b) 1 and 21 (c) 3 and 7 (d) 1 and 9
18. 18.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)
19. 19.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
20. 20.If a + b = 19 and ab = 88, what is a² + b²? (a) 273 (b) 361 (c) 537 (d) 185
21. 21.What are the roots of x² − 12x + 27 = 0? (a) 2 and 10 (b) 1 and 11 (c) 3 and 9 (d) 1 and 27
22. 22.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 425 (b) 225 (c) 625 (d) 635
23. 23.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 10816 (b) 9984 (c) 9216 (d) 9226
24. 24.What is the product of the roots of x² − 13x + 42 = 0? (a) 41 (b) 43 (c) 42 (d) 13
25. 25.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 1 and 63 (c) 7 and 9 (d) 3 and 21
26. 26.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 15 (c) 16 (d) 14
27. 27.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 11) (b) (x − 3)(x − 9) (c) (x + 3)(x + 9) (d) (x + 1)(x + 27)
28. 28.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x − 2)(x − 4) (c) (x + 2)(x + 4) (d) (x + 1)(x + 5)
29. 29.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
30. 30.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 1)(x + 16) (c) (x + 2)(x + 8) (d) (x − 2)(x − 8)
31. 31.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 213 (c) 147 (d) 125
32. 32.If a + b = 14 and ab = 45, what is (a − b)²? (a) 196 (b) 16 (c) 106 (d) 4
33. 33.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
34. 34.If a + b = 12 and ab = 32, what is a² + b²? (a) 80 (b) 144 (c) 112 (d) 208
35. 35.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
36. 36.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 73 (c) 121 (d) 97
37. 37.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 64 (c) 62 (d) 16
38. 38.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 8 (c) 256 (d) 160
39. 39.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 157 (c) 421 (d) 223
40. 40.If a + b = 4 and ab = 3, what is (a − b)²? (a) 16 (b) 2 (c) 10 (d) 4

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