Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 1 · 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 + 54? (a) (x + 1)(x + 54) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
2. 2.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)
3. 3.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 841 (b) 901 (c) 971 (d) 961
4. 4.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 1)(x + 8) (c) (x + 3)(x + 6) (d) (x + 1)(x + 18)
5. 5.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3844 (c) 3364 (d) 3604
6. 6.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 9) (c) (x + 4)(x + 6) (d) (x + 1)(x + 24)
7. 7.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 37 (c) 73 (d) 49
8. 8.If a + b = 19 and ab = 88, what is (a − b)²? (a) 361 (b) 3 (c) 9 (d) 185
9. 9.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 225 (c) 113 (d) 1
10. 10.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 6 (c) 1 and 12 (d) 2 and 6
11. 11.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x − 5)(x − 9)
12. 12.If a + b = 11 and ab = 24, what is (a − b)²? (a) 73 (b) 25 (c) 5 (d) 121
13. 13.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 11) (c) (x + 1)(x + 27) (d) (x + 3)(x + 9)
14. 14.If a + b = 10 and ab = 21, what is a² + b²? (a) 79 (b) 100 (c) 58 (d) 142
15. 15.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 1 (c) 289 (d) 217
16. 16.Using (a − b)² = a² − 2ab + b², what is 57²? (a) 3249 (b) 3259 (c) 3591 (d) 3969
17. 17.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 3136 (c) 4106 (d) 4096
18. 18.If a + b = 8 and ab = 15, what is (a − b)²? (a) 34 (b) 64 (c) 2 (d) 4
19. 19.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 28 (c) 12 (d) 26
20. 20.If a + b = 15 and ab = 36, what is (a − b)²? (a) 9 (b) 153 (c) 225 (d) 81
21. 21.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1379 (b) 1591 (c) 1849 (d) 1369
22. 22.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 16 (c) 324 (d) 170
23. 23.If a + b = 13 and ab = 22, what is (a − b)²? (a) 9 (b) 125 (c) 169 (d) 81
24. 24.What are the roots of x² − 9x + 18 = 0? (a) 3 and 6 (b) 1 and 8 (c) 2 and 9 (d) 1 and 18
25. 25.If a + b = 16 and ab = 63, what is (a − b)²? (a) 2 (b) 256 (c) 4 (d) 130
26. 26.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 6) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 2)(x + 5)
27. 27.If a + b = 14 and ab = 24, what is (a − b)²? (a) 148 (b) 100 (c) 196 (d) 10
28. 28.If a + b = 13 and ab = 40, what is a² + b²? (a) 249 (b) 169 (c) 89 (d) 129
29. 29.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2126 (b) 2116 (c) 2484 (d) 2916
30. 30.What are the roots of x² − 15x + 56 = 0? (a) 2 and 28 (b) 1 and 56 (c) 1 and 14 (d) 7 and 8
31. 31.If a + b = 11 and ab = 30, what is (a − b)²? (a) 121 (b) 91 (c) 1 (d) 61
32. 32.What is the sum of the roots of x² − 15x + 54 = 0? (a) 54 (b) 14 (c) 15 (d) 16
33. 33.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 1)(x + 4) (c) (x + 2)(x + 3) (d) (x − 2)(x − 3)
34. 34.Using (a − b)² = a² − 2ab + b², what is 96²? (a) 9226 (b) 9216 (c) 10816 (d) 9984
35. 35.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 784 (b) 1024 (c) 794 (d) 896
36. 36.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 235 (d) 625
37. 37.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
38. 38.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x − 2)(x − 7) (c) (x + 1)(x + 14) (d) (x + 1)(x + 8)
39. 39.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 127 (c) 169 (d) 253
40. 40.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 61 (c) 41 (d) 1

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