Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 380 · 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 = 5 and ab = 4, what is a² + b²? (a) 21 (b) 33 (c) 25 (d) 17
2. 2.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 371 (c) 361 (d) 399
3. 3.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 7744 (b) 8474 (c) 8464 (d) 8104
4. 4.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3364 (b) 3374 (c) 3596 (d) 3844
5. 5.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 9025 (b) 11035 (c) 11025 (d) 10025
6. 6.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 6 and 9 (c) 2 and 27 (d) 1 and 14
7. 7.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)
8. 8.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1946 (c) 1296 (d) 1936
9. 9.What is the product of the roots of x² − 15x + 56 = 0? (a) 15 (b) 55 (c) 56 (d) 57
10. 10.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 2 and 36 (d) 1 and 72
11. 11.If a + b = 19 and ab = 84, what is a² + b²? (a) 277 (b) 529 (c) 193 (d) 361
12. 12.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 217 (c) 289 (d) 145
13. 13.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 169 (c) 89 (d) 3
14. 14.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 100 (c) 292 (d) 148
15. 15.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)
16. 16.If a + b = 15 and ab = 54, what is (a − b)²? (a) 117 (b) 9 (c) 3 (d) 225
17. 17.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1591 (b) 1849 (c) 1369 (d) 1379
18. 18.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 1)(x + 54) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
19. 19.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 841 (c) 899 (d) 961
20. 20.If a + b = 13 and ab = 22, what is a² + b²? (a) 169 (b) 147 (c) 125 (d) 213
21. 21.What is the product of the roots of x² − 9x + 20 = 0? (a) 21 (b) 19 (c) 20 (d) 9
22. 22.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 256 (c) 266 (d) 576
23. 23.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 24) (b) (x + 3)(x + 8) (c) (x + 1)(x + 10) (d) (x − 3)(x − 8)
24. 24.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 361 (d) 271
25. 25.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 49 (c) 14 (d) 48
26. 26.What is the sum of the roots of x² − 13x + 36 = 0? (a) 14 (b) 12 (c) 13 (d) 36
27. 27.What is the product of the roots of x² − 7x + 10 = 0? (a) 7 (b) 9 (c) 11 (d) 10
28. 28.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7066 (b) 5776 (c) 6416 (d) 7056
29. 29.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 54 (c) 15 (d) 55
30. 30.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
31. 31.Which is the correct factorisation of x² + 10x + 24? (a) (x + 1)(x + 24) (b) (x − 4)(x − 6) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
32. 32.If a + b = 10 and ab = 24, what is (a − b)²? (a) 100 (b) 52 (c) 2 (d) 4
33. 33.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8109 (b) 8659 (c) 7569 (d) 8649
34. 34.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6889 (c) 6899 (d) 6409
35. 35.If a + b = 11 and ab = 28, what is (a − b)²? (a) 121 (b) 3 (c) 9 (d) 65
36. 36.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
37. 37.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 20)
38. 38.Using (a − b)² = a² − 2ab + b², what is 56²? (a) 3136 (b) 3146 (c) 3584 (d) 4096
39. 39.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 181 (d) 541
40. 40.What is the product of the roots of x² − 16x + 63 = 0? (a) 64 (b) 62 (c) 16 (d) 63

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