Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 97 · 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 = 15 and ab = 44, what is (a − b)²? (a) 137 (b) 225 (c) 49 (d) 7
2. 2.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 441 (b) 361 (c) 399 (d) 371
3. 3.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 14 (c) 45 (d) 15
4. 4.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 5041 (b) 4761 (c) 4771 (d) 4899
5. 5.What are the roots of x² − 10x + 24 = 0? (a) 1 and 9 (b) 4 and 6 (c) 2 and 12 (d) 1 and 24
6. 6.What is the product of the roots of x² − 9x + 14 = 0? (a) 14 (b) 9 (c) 15 (d) 13
7. 7.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1584 (b) 1306 (c) 1296 (d) 1936
8. 8.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3731 (b) 3481 (c) 3601 (d) 3721
9. 9.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 13) (c) (x + 1)(x + 45) (d) (x + 5)(x + 9)
10. 10.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 72 (d) 17
11. 11.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1681 (b) 1599 (c) 1521 (d) 1531
12. 12.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 2)(x + 3) (c) (x + 1)(x + 4) (d) (x + 1)(x + 6)
13. 13.If a + b = 6 and ab = 8, what is (a − b)²? (a) 20 (b) 4 (c) 2 (d) 36
14. 14.If a + b = 7 and ab = 12, what is a² + b²? (a) 25 (b) 37 (c) 49 (d) 73
15. 15.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
16. 16.If a + b = 3 and ab = 2, what is (a − b)²? (a) 5 (b) 1 (c) 9 (d) 7
17. 17.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 164 (c) 324 (d) 484
18. 18.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 5486 (b) 5476 (c) 4356 (d) 4916
19. 19.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 72 (d) 73
20. 20.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7225 (b) 5625 (c) 7235 (d) 6425
21. 21.If a + b = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 9 (d) 3
22. 22.What are the roots of x² − 11x + 24 = 0? (a) 2 and 12 (b) 3 and 8 (c) 1 and 10 (d) 1 and 24
23. 23.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 3)(x + 5) (c) (x + 1)(x + 7) (d) (x − 3)(x − 5)
24. 24.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)
25. 25.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 271 (d) 181
26. 26.If a + b = 8 and ab = 7, what is (a − b)²? (a) 6 (b) 50 (c) 64 (d) 36
27. 27.If a + b = 12 and ab = 27, what is (a − b)²? (a) 36 (b) 144 (c) 6 (d) 90
28. 28.If a + b = 10 and ab = 16, what is a² + b²? (a) 100 (b) 68 (c) 84 (d) 132
29. 29.If a + b = 21 and ab = 108, what is a² + b²? (a) 333 (b) 441 (c) 657 (d) 225
30. 30.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 5929 (c) 6889 (d) 6899
31. 31.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2496 (b) 2704 (c) 2314 (d) 2304
32. 32.If a + b = 5 and ab = 4, what is a² + b²? (a) 25 (b) 21 (c) 33 (d) 17
33. 33.What is the product of the roots of x² − 10x + 21 = 0? (a) 20 (b) 21 (c) 10 (d) 22
34. 34.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x − 8)(x − 9) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
35. 35.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 256 (b) 416 (c) 576 (d) 586
36. 36.What is the sum of the roots of x² − 11x + 24 = 0? (a) 12 (b) 10 (c) 24 (d) 11
37. 37.Which is the correct factorisation of x² + 6x + 8? (a) (x + 1)(x + 8) (b) (x + 1)(x + 5) (c) (x + 2)(x + 4) (d) (x − 2)(x − 4)
38. 38.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 11025 (c) 10025 (d) 9025
39. 39.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5635 (b) 5625 (c) 4925 (d) 4225
40. 40.What is the sum of the roots of x² − 9x + 20 = 0? (a) 8 (b) 9 (c) 20 (d) 10

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