Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 460 · 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 product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 48 (c) 47 (d) 49
2. 2.What is the sum of the roots of x² − 10x + 24 = 0? (a) 24 (b) 9 (c) 11 (d) 10
3. 3.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 54 (c) 15 (d) 53
4. 4.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
5. 5.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 40) (b) (x − 5)(x − 8) (c) (x + 1)(x + 12) (d) (x + 5)(x + 8)
6. 6.If a + b = 12 and ab = 11, what is (a − b)²? (a) 122 (b) 144 (c) 100 (d) 10
7. 7.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 229 (c) 139 (d) 109
8. 8.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 117 (c) 333 (d) 225
9. 9.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 896 (b) 794 (c) 784 (d) 1024
10. 10.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 592 (d) 304
11. 11.If a + b = 13 and ab = 22, what is a² + b²? (a) 147 (b) 213 (c) 125 (d) 169
12. 12.If a + b = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 100 (c) 68 (d) 36
13. 13.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 6)
14. 14.If a + b = 13 and ab = 40, what is (a − b)²? (a) 169 (b) 9 (c) 89 (d) 3
15. 15.What are the roots of x² − 7x + 10 = 0? (a) 1 and 10 (b) 3 and 4 (c) 1 and 6 (d) 2 and 5
16. 16.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x − 6)(x − 8) (c) (x + 6)(x + 8) (d) (x + 1)(x + 13)
17. 17.If a + b = 6 and ab = 8, what is (a − b)²? (a) 4 (b) 36 (c) 2 (d) 20
18. 18.If a + b = 19 and ab = 88, what is a² + b²? (a) 361 (b) 185 (c) 537 (d) 273
19. 19.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 10) (c) (x − 5)(x − 6) (d) (x + 1)(x + 30)
20. 20.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 7396 (c) 8836 (d) 8084
21. 21.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 3 and 21 (c) 1 and 15 (d) 1 and 63
22. 22.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x − 8)(x − 9) (d) (x + 8)(x + 9)
23. 23.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 127 (c) 85 (d) 169
24. 24.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1859 (c) 1849 (d) 1609
25. 25.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)
26. 26.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 217 (c) 145 (d) 1
27. 27.Using (a − b)² = a² − 2ab + b², what is 67²? (a) 5329 (b) 4891 (c) 4489 (d) 4499
28. 28.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x − 5)(x − 9)
29. 29.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 169 (c) 225 (d) 337
30. 30.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 361 (c) 401 (d) 451
31. 31.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1369 (b) 1591 (c) 1849 (d) 1379
32. 32.If a + b = 12 and ab = 27, what is a² + b²? (a) 144 (b) 90 (c) 117 (d) 198
33. 33.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 64 (c) 8 (d) 100
34. 34.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
35. 35.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 7056 (b) 5786 (c) 6384 (d) 5776
36. 36.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9811 (c) 9801 (d) 9999
37. 37.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x + 1)(x + 14)
38. 38.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3844 (b) 3604 (c) 3854 (d) 3364
39. 39.What are the roots of x² − 14x + 48 = 0? (a) 1 and 48 (b) 1 and 13 (c) 6 and 8 (d) 2 and 24
40. 40.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 145 (d) 289

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