Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 965 · 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.If a + b = 7 and ab = 12, what is (a − b)²? (a) 25 (b) 49 (c) 37 (d) 1
2. 2.If a + b = 15 and ab = 54, what is a² + b²? (a) 333 (b) 171 (c) 225 (d) 117
3. 3.If a + b = 15 and ab = 44, what is (a − b)²? (a) 225 (b) 7 (c) 137 (d) 49
4. 4.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1601 (b) 1681 (c) 1691 (d) 1521
5. 5.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1609 (b) 1859 (c) 1369 (d) 1849
6. 6.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x − 3)(x − 8) (c) (x + 1)(x + 24) (d) (x + 3)(x + 8)
7. 7.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6561 (c) 6401 (d) 6241
8. 8.If a + b = 12 and ab = 11, what is a² + b²? (a) 166 (b) 122 (c) 144 (d) 133
9. 9.What is the sum of the roots of x² − 13x + 42 = 0? (a) 14 (b) 42 (c) 12 (d) 13
10. 10.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 36 (c) 12 (d) 35
11. 11.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 884 (b) 1156 (c) 686 (d) 676
12. 12.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 130 (c) 4 (d) 2
13. 13.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9811 (b) 10201 (c) 9801 (d) 9999
14. 14.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 225 (c) 3 (d) 9
15. 15.Which is the correct factorisation of x² + 9x + 14? (a) (x − 2)(x − 7) (b) (x + 2)(x + 7) (c) (x + 1)(x + 8) (d) (x + 1)(x + 14)
16. 16.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 25 (c) 49 (d) 73
17. 17.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x − 8)(x − 9) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
18. 18.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 289 (c) 391 (d) 299
19. 19.If a + b = 6 and ab = 5, what is (a − b)²? (a) 36 (b) 26 (c) 16 (d) 4
20. 20.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 7)(x + 9) (c) (x + 1)(x + 63) (d) (x + 1)(x + 15)
21. 21.If a + b = 17 and ab = 70, what is (a − b)²? (a) 289 (b) 3 (c) 9 (d) 149
22. 22.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 7)(x + 8) (d) (x + 1)(x + 56)
23. 23.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 17 (d) 72
24. 24.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 1 and 32 (c) 2 and 16 (d) 1 and 11
25. 25.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 1)(x + 14)
26. 26.What is the product of the roots of x² − 5x + 6 = 0? (a) 5 (b) 8 (c) 6 (d) 7
27. 27.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 23 (c) 25 (d) 24
28. 28.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
29. 29.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 36 (c) 180 (d) 6
30. 30.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 5476 (c) 4366 (d) 4356
31. 31.What is the sum of the roots of x² − 9x + 14 = 0? (a) 9 (b) 10 (c) 14 (d) 8
32. 32.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2496 (d) 2304
33. 33.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 18 (c) 10 (d) 11
34. 34.What are the roots of x² − 15x + 54 = 0? (a) 2 and 27 (b) 6 and 9 (c) 1 and 54 (d) 1 and 14
35. 35.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 5 (c) 225 (d) 125
36. 36.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x − 2)(x − 9) (c) (x + 1)(x + 10) (d) (x + 2)(x + 9)
37. 37.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 169 (c) 3 (d) 89
38. 38.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3374 (c) 3596 (d) 3844
39. 39.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4624 (c) 5184 (d) 4634
40. 40.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 11) (b) (x + 1)(x + 35) (c) (x − 5)(x − 7) (d) (x + 5)(x + 7)

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