Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 256 · 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 = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 170 (c) 16 (d) 324
2. 2.Which is the correct factorisation of x² + 10x + 24? (a) (x − 4)(x − 6) (b) (x + 1)(x + 24) (c) (x + 1)(x + 9) (d) (x + 4)(x + 6)
3. 3.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 18 (c) 16 (d) 17
4. 4.What are the roots of x² − 14x + 45 = 0? (a) 5 and 9 (b) 1 and 45 (c) 1 and 13 (d) 3 and 15
5. 5.What is the product of the roots of x² − 12x + 35 = 0? (a) 34 (b) 36 (c) 12 (d) 35
6. 6.If a + b = 8 and ab = 12, what is a² + b²? (a) 52 (b) 88 (c) 40 (d) 64
7. 7.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
8. 8.If a + b = 14 and ab = 48, what is (a − b)²? (a) 196 (b) 4 (c) 100 (d) 2
9. 9.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1936 (b) 1616 (c) 1296 (d) 1946
10. 10.If a + b = 18 and ab = 72, what is (a − b)²? (a) 180 (b) 324 (c) 36 (d) 6
11. 11.Using (a + b)² = a² + 2ab + b², what is 71²? (a) 5051 (b) 5041 (c) 4761 (d) 4901
12. 12.What is the product of the roots of x² − 15x + 56 = 0? (a) 55 (b) 57 (c) 56 (d) 15
13. 13.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 10211 (b) 10201 (c) 9801 (d) 10001
14. 14.Which is the correct factorisation of x² + 14x + 45? (a) (x + 1)(x + 45) (b) (x + 5)(x + 9) (c) (x + 1)(x + 13) (d) (x − 5)(x − 9)
15. 15.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 1 and 36 (c) 4 and 9 (d) 2 and 18
16. 16.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1444 (c) 1764 (d) 1774
17. 17.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 361 (b) 399 (c) 371 (d) 441
18. 18.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)
19. 19.If a + b = 13 and ab = 40, what is a² + b²? (a) 169 (b) 249 (c) 129 (d) 89
20. 20.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4634 (b) 4896 (c) 5184 (d) 4624
21. 21.If a + b = 16 and ab = 55, what is a² + b²? (a) 146 (b) 256 (c) 366 (d) 201
22. 22.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 11 (c) 27 (d) 13
23. 23.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2304 (b) 2714 (c) 2504 (d) 2704
24. 24.Using (a − b)² = a² − 2ab + b², what is 58²? (a) 3844 (b) 3596 (c) 3374 (d) 3364
25. 25.Using (a + b)² = a² + 2ab + b², what is 31²? (a) 901 (b) 841 (c) 971 (d) 961
26. 26.What is the product of the roots of x² − 5x + 6 = 0? (a) 8 (b) 7 (c) 5 (d) 6
27. 27.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9991 (c) 10609 (d) 9409
28. 28.If a + b = 18 and ab = 80, what is a² + b²? (a) 244 (b) 324 (c) 484 (d) 164
29. 29.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x + 1)(x + 30) (c) (x + 1)(x + 10) (d) (x − 5)(x − 6)
30. 30.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 1 and 12 (d) 2 and 21
31. 31.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3854 (b) 3364 (c) 3844 (d) 3604
32. 32.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x + 1)(x + 28) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
33. 33.What are the roots of x² − 10x + 16 = 0? (a) 1 and 9 (b) 1 and 16 (c) 2 and 8 (d) 3 and 7
34. 34.What is the product of the roots of x² − 6x + 8 = 0? (a) 7 (b) 9 (c) 6 (d) 8
35. 35.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
36. 36.What are the roots of x² − 9x + 20 = 0? (a) 4 and 5 (b) 1 and 8 (c) 2 and 10 (d) 1 and 20
37. 37.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 2)(x + 9) (c) (x + 1)(x + 10) (d) (x − 2)(x − 9)
38. 38.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 404 (b) 324 (c) 484 (d) 494
39. 39.Which is the correct factorisation of x² + 9x + 14? (a) (x + 2)(x + 7) (b) (x + 1)(x + 8) (c) (x − 2)(x − 7) (d) (x + 1)(x + 14)
40. 40.Which is the correct factorisation of x² + 13x + 36? (a) (x + 1)(x + 36) (b) (x + 1)(x + 12) (c) (x − 4)(x − 9) (d) (x + 4)(x + 9)

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