Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 264 · 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.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6889 (b) 5929 (c) 6391 (d) 5939
2. 2.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 416 (b) 256 (c) 576 (d) 586
3. 3.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 1 and 30 (c) 2 and 15 (d) 5 and 6
4. 4.Which is the correct factorisation of x² + 12x + 32? (a) (x + 4)(x + 8) (b) (x − 4)(x − 8) (c) (x + 1)(x + 32) (d) (x + 1)(x + 11)
5. 5.If a + b = 5 and ab = 4, what is (a − b)²? (a) 9 (b) 25 (c) 17 (d) 3
6. 6.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 54) (b) (x + 1)(x + 14) (c) (x − 6)(x − 9) (d) (x + 6)(x + 9)
7. 7.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
8. 8.Which is the correct factorisation of x² + 14x + 45? (a) (x + 5)(x + 9) (b) (x + 1)(x + 13) (c) (x − 5)(x − 9) (d) (x + 1)(x + 45)
9. 9.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 576 (c) 256 (d) 266
10. 10.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 401 (b) 361 (c) 451 (d) 441
11. 11.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 11025 (b) 9035 (c) 9025 (d) 9975
12. 12.Using (a − b)² = a² − 2ab + b², what is 59²? (a) 3599 (b) 3491 (c) 3481 (d) 3721
13. 13.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 4 (c) 2 (d) 202
14. 14.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 72 (d) 18
15. 15.If a + b = 18 and ab = 80, what is (a − b)²? (a) 324 (b) 4 (c) 164 (d) 2
16. 16.Which is the correct factorisation of x² + 12x + 35? (a) (x + 5)(x + 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x − 5)(x − 7)
17. 17.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 15 (c) 63 (d) 17
18. 18.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 333 (d) 441
19. 19.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10016 (b) 10826 (c) 10816 (d) 9216
20. 20.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 13 (c) 12 (d) 27
21. 21.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 271 (d) 1
22. 22.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8474 (b) 7744 (c) 8104 (d) 8464
23. 23.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 169 (c) 225 (d) 337
24. 24.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1225 (b) 625 (c) 1235 (d) 925
25. 25.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 7 and 9 (c) 1 and 63 (d) 1 and 15
26. 26.If a + b = 17 and ab = 72, what is (a − b)²? (a) 289 (b) 1 (c) 217 (d) 145
27. 27.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1681 (c) 1691 (d) 1601
28. 28.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 201 (c) 366 (d) 256
29. 29.If a + b = 16 and ab = 55, what is (a − b)²? (a) 146 (b) 256 (c) 6 (d) 36
30. 30.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1099 (c) 909 (d) 1089
31. 31.What is the product of the roots of x² − 15x + 54 = 0? (a) 55 (b) 15 (c) 53 (d) 54
32. 32.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 52 (c) 148 (d) 100
33. 33.What is the product of the roots of x² − 10x + 24 = 0? (a) 10 (b) 25 (c) 23 (d) 24
34. 34.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 6 and 9 (c) 1 and 14 (d) 2 and 27
35. 35.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 7235 (c) 8075 (d) 9025
36. 36.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 170 (c) 478 (d) 324
37. 37.What is the sum of the roots of x² − 11x + 24 = 0? (a) 24 (b) 11 (c) 12 (d) 10
38. 38.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 7) (c) (x + 1)(x + 15) (d) (x + 3)(x + 5)
39. 39.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 24 (c) 23 (d) 11
40. 40.What is the product of the roots of x² − 9x + 18 = 0? (a) 17 (b) 18 (c) 9 (d) 19

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