Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 767 · 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 46²? (a) 2916 (b) 2126 (c) 2484 (d) 2116
2. 2.If a + b = 19 and ab = 90, what is (a − b)²? (a) 271 (b) 1 (c) 181 (d) 361
3. 3.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 10609 (c) 9419 (d) 9409
4. 4.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x + 1)(x + 6) (c) (x + 2)(x + 5) (d) (x − 2)(x − 5)
5. 5.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
6. 6.If a + b = 19 and ab = 84, what is a² + b²? (a) 361 (b) 529 (c) 277 (d) 193
7. 7.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 2 and 5 (c) 1 and 8 (d) 2 and 4
8. 8.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2496 (c) 2704 (d) 2314
9. 9.If a + b = 14 and ab = 48, what is a² + b²? (a) 196 (b) 292 (c) 100 (d) 148
10. 10.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
11. 11.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 1)(x + 56) (c) (x + 7)(x + 8) (d) (x − 7)(x − 8)
12. 12.Which is the correct factorisation of x² + 10x + 16? (a) (x − 2)(x − 8) (b) (x + 2)(x + 8) (c) (x + 1)(x + 9) (d) (x + 1)(x + 16)
13. 13.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x − 6)(x − 7) (c) (x + 1)(x + 12) (d) (x + 1)(x + 42)
14. 14.What is the sum of the roots of x² − 8x + 12 = 0? (a) 12 (b) 8 (c) 9 (d) 7
15. 15.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7225 (c) 5625 (d) 7235
16. 16.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 9025 (b) 8125 (c) 7225 (d) 9035
17. 17.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 3)(x + 4) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
18. 18.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6084 (b) 6396 (c) 6724 (d) 6094
19. 19.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 10 (c) 8 (d) 20
20. 20.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 54) (c) (x + 1)(x + 14) (d) (x + 6)(x + 9)
21. 21.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 130 (c) 4 (d) 2
22. 22.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10609 (b) 9409 (c) 10619 (d) 10009
23. 23.If a + b = 18 and ab = 72, what is a² + b²? (a) 468 (b) 324 (c) 252 (d) 180
24. 24.If a + b = 14 and ab = 45, what is a² + b²? (a) 286 (b) 196 (c) 151 (d) 106
25. 25.If a + b = 21 and ab = 108, what is a² + b²? (a) 441 (b) 657 (c) 333 (d) 225
26. 26.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 72 (c) 17 (d) 16
27. 27.What are the roots of x² − 10x + 24 = 0? (a) 1 and 24 (b) 1 and 9 (c) 4 and 6 (d) 2 and 12
28. 28.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x − 6)(x − 8)
29. 29.If a + b = 5 and ab = 6, what is (a − b)²? (a) 19 (b) 13 (c) 1 (d) 25
30. 30.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 299 (c) 391 (d) 289
31. 31.What are the roots of x² − 10x + 21 = 0? (a) 1 and 9 (b) 1 and 21 (c) 3 and 7 (d) 2 and 8
32. 32.If a + b = 13 and ab = 36, what is (a − b)²? (a) 25 (b) 97 (c) 169 (d) 5
33. 33.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 14) (b) (x − 2)(x − 7) (c) (x + 2)(x + 7) (d) (x + 1)(x + 8)
34. 34.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1531 (c) 1599 (d) 1521
35. 35.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 49 (c) 37 (d) 25
36. 36.What is the sum of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 14 (d) 16
37. 37.What is the sum of the roots of x² − 15x + 54 = 0? (a) 16 (b) 15 (c) 54 (d) 14
38. 38.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
39. 39.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 451 (b) 441 (c) 401 (d) 361
40. 40.What is the product of the roots of x² − 7x + 12 = 0? (a) 12 (b) 7 (c) 13 (d) 11

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