Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 565 · 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 = 50, what is (a − b)²? (a) 225 (b) 5 (c) 125 (d) 25
2. 2.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4634 (c) 4624 (d) 5184
3. 3.If a + b = 11 and ab = 28, what is a² + b²? (a) 121 (b) 93 (c) 177 (d) 65
4. 4.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
5. 5.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2516 (b) 2116 (c) 2926 (d) 2916
6. 6.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1369 (c) 1591 (d) 1379
7. 7.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6251 (c) 6399 (d) 6561
8. 8.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 5929 (c) 6391 (d) 6889
9. 9.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6899 (b) 6409 (c) 5929 (d) 6889
10. 10.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 12 (c) 13 (d) 35
11. 11.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 9 (c) 225 (d) 153
12. 12.If a + b = 15 and ab = 54, what is a² + b²? (a) 171 (b) 117 (c) 225 (d) 333
13. 13.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3481 (b) 3601 (c) 3721 (d) 3731
14. 14.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 400 (c) 16 (d) 4
15. 15.What is the sum of the roots of x² − 10x + 16 = 0? (a) 11 (b) 16 (c) 10 (d) 9
16. 16.Using (a + b)² = a² + 2ab + b², what is 95²? (a) 8125 (b) 7225 (c) 9025 (d) 9035
17. 17.If a + b = 10 and ab = 16, what is (a − b)²? (a) 100 (b) 36 (c) 6 (d) 68
18. 18.If a + b = 16 and ab = 63, what is (a − b)²? (a) 256 (b) 2 (c) 130 (d) 4
19. 19.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 235 (b) 625 (c) 225 (d) 375
20. 20.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
21. 21.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1764 (c) 1444 (d) 1454
22. 22.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 10201 (b) 9811 (c) 9999 (d) 9801
23. 23.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 6 and 9 (c) 2 and 27 (d) 1 and 14
24. 24.If a + b = 12 and ab = 35, what is (a − b)²? (a) 144 (b) 74 (c) 4 (d) 2
25. 25.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)
26. 26.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 8)(x + 9) (c) (x + 1)(x + 16) (d) (x + 1)(x + 72)
27. 27.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 1 and 72 (d) 8 and 9
28. 28.If a + b = 11 and ab = 10, what is (a − b)²? (a) 9 (b) 121 (c) 101 (d) 81
29. 29.What are the roots of x² − 16x + 63 = 0? (a) 3 and 21 (b) 1 and 15 (c) 1 and 63 (d) 7 and 9
30. 30.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 3025 (b) 2525 (c) 2025 (d) 3035
31. 31.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11035 (b) 11025 (c) 10025 (d) 9025
32. 32.What is the product of the roots of x² − 15x + 54 = 0? (a) 15 (b) 53 (c) 55 (d) 54
33. 33.Which is the correct factorisation of x² + 12x + 35? (a) (x + 1)(x + 35) (b) (x + 1)(x + 11) (c) (x + 5)(x + 7) (d) (x − 5)(x − 7)
34. 34.If a + b = 18 and ab = 72, what is a² + b²? (a) 180 (b) 324 (c) 468 (d) 252
35. 35.What is the product of the roots of x² − 12x + 32 = 0? (a) 12 (b) 31 (c) 32 (d) 33
36. 36.What is the product of the roots of x² − 12x + 27 = 0? (a) 27 (b) 28 (c) 12 (d) 26
37. 37.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 225 (c) 325 (d) 125
38. 38.Using (a + b)² = a² + 2ab + b², what is 102²? (a) 10414 (b) 10004 (c) 9604 (d) 10404
39. 39.Which is the correct factorisation of x² + 12x + 27? (a) (x + 1)(x + 27) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x − 3)(x − 9)
40. 40.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1681 (c) 1521 (d) 1601

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