Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 971 · 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 = 8 and ab = 7, what is a² + b²? (a) 64 (b) 78 (c) 50 (d) 57
2. 2.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
3. 3.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10004 (b) 10404 (c) 9604 (d) 10414
4. 4.If a + b = 19 and ab = 84, what is (a − b)²? (a) 5 (b) 361 (c) 25 (d) 193
5. 5.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 16 (c) 4 (d) 324
6. 6.Using (a + b)² = a² + 2ab + b², what is 92²? (a) 8104 (b) 7744 (c) 8464 (d) 8474
7. 7.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)
8. 8.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 1235 (b) 625 (c) 925 (d) 1225
9. 9.If a + b = 11 and ab = 28, what is a² + b²? (a) 93 (b) 121 (c) 65 (d) 177
10. 10.If a + b = 17 and ab = 72, what is a² + b²? (a) 433 (b) 289 (c) 217 (d) 145
11. 11.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)
12. 12.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 10 (c) 12 (d) 11
13. 13.What is the product of the roots of x² − 15x + 54 = 0? (a) 53 (b) 55 (c) 54 (d) 15
14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x − 7)(x − 8) (c) (x + 7)(x + 8) (d) (x + 1)(x + 14)
15. 15.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 125 (c) 325 (d) 225
16. 16.If a + b = 13 and ab = 40, what is (a − b)²? (a) 9 (b) 3 (c) 89 (d) 169
17. 17.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1235 (b) 1225 (c) 2025 (d) 1575
18. 18.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 441 (c) 333 (d) 657
19. 19.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 4356 (c) 5476 (d) 4916
20. 20.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 6425 (b) 7235 (c) 7225 (d) 5625
21. 21.What are the roots of x² − 12x + 27 = 0? (a) 3 and 9 (b) 1 and 11 (c) 2 and 10 (d) 1 and 27
22. 22.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2411 (b) 2401 (c) 2601 (d) 2499
23. 23.What is the sum of the roots of x² − 5x + 6 = 0? (a) 5 (b) 4 (c) 7 (d) 6
24. 24.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 1)(x + 63) (d) (x + 7)(x + 9)
25. 25.What is the sum of the roots of x² − 14x + 48 = 0? (a) 48 (b) 14 (c) 13 (d) 15
26. 26.Which is the correct factorisation of x² + 9x + 14? (a) (x + 1)(x + 8) (b) (x + 1)(x + 14) (c) (x − 2)(x − 7) (d) (x + 2)(x + 7)
27. 27.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 891 (b) 1089 (c) 729 (d) 739
28. 28.What is the product of the roots of x² − 14x + 48 = 0? (a) 14 (b) 47 (c) 48 (d) 49
29. 29.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)
30. 30.Using (a − b)² = a² − 2ab + b², what is 87²? (a) 8091 (b) 8649 (c) 7569 (d) 7579
31. 31.Which is the correct factorisation of x² + 9x + 18? (a) (x + 1)(x + 8) (b) (x + 3)(x + 6) (c) (x + 1)(x + 18) (d) (x − 3)(x − 6)
32. 32.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 409 (c) 539 (d) 529
33. 33.Which is the correct factorisation of x² + 11x + 18? (a) (x + 2)(x + 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x − 2)(x − 9)
34. 34.If a + b = 10 and ab = 24, what is (a − b)²? (a) 2 (b) 52 (c) 4 (d) 100
35. 35.Which is the correct factorisation of x² + 15x + 54? (a) (x + 1)(x + 14) (b) (x + 6)(x + 9) (c) (x + 1)(x + 54) (d) (x − 6)(x − 9)
36. 36.Using (a + b)² = a² + 2ab + b², what is 43²? (a) 1369 (b) 1609 (c) 1859 (d) 1849
37. 37.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 219 (c) 149 (d) 429
38. 38.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 4 (c) 26 (d) 36
39. 39.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 5)(x + 8) (d) (x + 1)(x + 40)
40. 40.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 371 (c) 361 (d) 399

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