Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 269 · 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 + 14) (b) (x + 6)(x + 9) (c) (x − 6)(x − 9) (d) (x + 1)(x + 54)
2. 2.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 13) (b) (x − 5)(x − 9) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
3. 3.What are the roots of x² − 11x + 30 = 0? (a) 1 and 30 (b) 1 and 10 (c) 2 and 15 (d) 5 and 6
4. 4.What is the product of the roots of x² − 16x + 63 = 0? (a) 63 (b) 62 (c) 16 (d) 64
5. 5.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 273 (c) 537 (d) 185
6. 6.Which is the correct factorisation of x² + 12x + 32? (a) (x − 4)(x − 8) (b) (x + 1)(x + 11) (c) (x + 4)(x + 8) (d) (x + 1)(x + 32)
7. 7.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 384 (c) 576 (d) 256
8. 8.If a + b = 12 and ab = 11, what is (a − b)²? (a) 100 (b) 10 (c) 144 (d) 122
9. 9.What is the product of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 8 (d) 5
10. 10.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 11 (c) 12 (d) 35
11. 11.What are the roots of x² − 8x + 12 = 0? (a) 3 and 4 (b) 1 and 7 (c) 1 and 12 (d) 2 and 6
12. 12.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5929 (c) 5939 (d) 6391
13. 13.If a + b = 18 and ab = 77, what is a² + b²? (a) 324 (b) 247 (c) 478 (d) 170
14. 14.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 1089 (c) 729 (d) 739
15. 15.Which is the correct factorisation of x² + 11x + 24? (a) (x − 3)(x − 8) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x + 1)(x + 10)
16. 16.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 9 (c) 20 (d) 10
17. 17.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
18. 18.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 171 (c) 333 (d) 117
19. 19.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 3 (c) 117 (d) 9
20. 20.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2916 (b) 2926 (c) 2116 (d) 2516
21. 21.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 72 (c) 71 (d) 17
22. 22.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 74 (c) 144 (d) 109
23. 23.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1774 (c) 1764 (d) 1444
24. 24.If a + b = 16 and ab = 48, what is (a − b)²? (a) 64 (b) 256 (c) 160 (d) 8
25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
26. 26.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 1156 (b) 916 (c) 676 (d) 1166
27. 27.What is the product of the roots of x² − 11x + 30 = 0? (a) 31 (b) 11 (c) 30 (d) 29
28. 28.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)
29. 29.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)
30. 30.If a + b = 13 and ab = 42, what is a² + b²? (a) 127 (b) 85 (c) 169 (d) 253
31. 31.If a + b = 15 and ab = 44, what is (a − b)²? (a) 7 (b) 225 (c) 137 (d) 49
32. 32.What is the product of the roots of x² − 13x + 42 = 0? (a) 13 (b) 41 (c) 42 (d) 43
33. 33.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x − 7)(x − 8) (d) (x + 1)(x + 56)
34. 34.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 6 and 8 (c) 2 and 24 (d) 1 and 48
35. 35.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9035 (b) 9025 (c) 8125 (d) 7225
36. 36.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x − 2)(x − 3)
37. 37.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 1)(x + 12) (c) (x + 3)(x + 4) (d) (x − 3)(x − 4)
38. 38.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 375 (b) 225 (c) 235 (d) 625
39. 39.Using (a + b)² = a² + 2ab + b², what is 51²? (a) 2501 (b) 2401 (c) 2601 (d) 2611
40. 40.If a + b = 7 and ab = 10, what is (a − b)²? (a) 29 (b) 9 (c) 3 (d) 49

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