Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 307 · 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 47²? (a) 2491 (b) 2809 (c) 2219 (d) 2209
2. 2.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
3. 3.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)
4. 4.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2484 (c) 2116 (d) 2126
5. 5.If a + b = 12 and ab = 35, what is a² + b²? (a) 74 (b) 144 (c) 109 (d) 214
6. 6.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 3)(x + 5) (c) (x − 3)(x − 5) (d) (x + 1)(x + 7)
7. 7.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 361 (c) 441 (d) 371
8. 8.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 6391 (b) 5929 (c) 6889 (d) 5939
9. 9.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)
10. 10.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 539 (b) 409 (c) 529 (d) 289
11. 11.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 3 and 21 (c) 1 and 63 (d) 7 and 9
12. 12.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 739 (c) 729 (d) 1089
13. 13.If a + b = 8 and ab = 12, what is (a − b)²? (a) 64 (b) 40 (c) 4 (d) 16
14. 14.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 4 (c) 16 (d) 170
15. 15.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 425 (c) 625 (d) 635
16. 16.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3854 (c) 3364 (d) 3844
17. 17.Which is the correct factorisation of x² + 13x + 42? (a) (x + 6)(x + 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x − 6)(x − 7)
18. 18.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 334 (b) 484 (c) 396 (d) 324
19. 19.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 5)(x + 8) (c) (x + 1)(x + 40) (d) (x + 1)(x + 12)
20. 20.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 9 (c) 441 (d) 3
21. 21.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 56 (d) 16
22. 22.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 729 (b) 1089 (c) 1099 (d) 909
23. 23.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2304 (b) 2314 (c) 2704 (d) 2496
24. 24.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 14 (d) 12
25. 25.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
26. 26.What is the sum of the roots of x² − 7x + 12 = 0? (a) 12 (b) 6 (c) 7 (d) 8
27. 27.If a + b = 15 and ab = 54, what is a² + b²? (a) 225 (b) 117 (c) 333 (d) 171
28. 28.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 8096 (c) 7744 (d) 7754
29. 29.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 266 (b) 576 (c) 384 (d) 256
30. 30.If a + b = 5 and ab = 6, what is (a − b)²? (a) 25 (b) 13 (c) 19 (d) 1
31. 31.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x − 4)(x − 7)
32. 32.Which is the correct factorisation of x² + 15x + 54? (a) (x − 6)(x − 9) (b) (x + 1)(x + 14) (c) (x + 6)(x + 9) (d) (x + 1)(x + 54)
33. 33.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 10 (c) 2 (d) 16
34. 34.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 18) (b) (x − 3)(x − 6) (c) (x + 3)(x + 6) (d) (x + 1)(x + 8)
35. 35.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 304 (d) 592
36. 36.If a + b = 19 and ab = 90, what is a² + b²? (a) 181 (b) 271 (c) 361 (d) 541
37. 37.Which is the correct factorisation of x² + 14x + 48? (a) (x + 1)(x + 48) (b) (x + 1)(x + 13) (c) (x + 6)(x + 8) (d) (x − 6)(x − 8)
38. 38.Which is the correct factorisation of x² + 11x + 24? (a) (x + 1)(x + 10) (b) (x + 1)(x + 24) (c) (x + 3)(x + 8) (d) (x − 3)(x − 8)
39. 39.What are the roots of x² − 6x + 8 = 0? (a) 1 and 5 (b) 1 and 8 (c) 2 and 4 (d) 2 and 5
40. 40.What is the sum of the roots of x² − 13x + 36 = 0? (a) 12 (b) 13 (c) 14 (d) 36

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