Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 445 · 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 67²? (a) 4891 (b) 4499 (c) 4489 (d) 5329
2. 2.If a + b = 7 and ab = 12, what is a² + b²? (a) 37 (b) 73 (c) 25 (d) 49
3. 3.Using (a − b)² = a² − 2ab + b², what is 76²? (a) 5786 (b) 6384 (c) 5776 (d) 7056
4. 4.What are the roots of x² − 15x + 54 = 0? (a) 6 and 9 (b) 1 and 54 (c) 1 and 14 (d) 2 and 27
5. 5.If a + b = 15 and ab = 50, what is (a − b)²? (a) 25 (b) 5 (c) 125 (d) 225
6. 6.Which is the correct factorisation of x² + 11x + 30? (a) (x + 5)(x + 6) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 1)(x + 10)
7. 7.If a + b = 15 and ab = 56, what is a² + b²? (a) 113 (b) 169 (c) 337 (d) 225
8. 8.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 56) (b) (x + 1)(x + 14) (c) (x − 7)(x − 8) (d) (x + 7)(x + 8)
9. 9.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 7921 (b) 8281 (c) 8291 (d) 8101
10. 10.What are the roots of x² − 17x + 72 = 0? (a) 8 and 9 (b) 1 and 16 (c) 1 and 72 (d) 2 and 36
11. 11.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 20) (b) (x − 4)(x − 5) (c) (x + 4)(x + 5) (d) (x + 1)(x + 8)
12. 12.If a + b = 7 and ab = 6, what is (a − b)²? (a) 49 (b) 5 (c) 37 (d) 25
13. 13.Which is the correct factorisation of x² + 12x + 32? (a) (x + 1)(x + 32) (b) (x + 1)(x + 11) (c) (x − 4)(x − 8) (d) (x + 4)(x + 8)
14. 14.If a + b = 6 and ab = 5, what is a² + b²? (a) 31 (b) 36 (c) 46 (d) 26
15. 15.What are the roots of x² − 10x + 24 = 0? (a) 2 and 12 (b) 1 and 9 (c) 4 and 6 (d) 1 and 24
16. 16.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10816 (c) 10016 (d) 10826
17. 17.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)
18. 18.What is the product of the roots of x² − 9x + 20 = 0? (a) 19 (b) 21 (c) 20 (d) 9
19. 19.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 17 (c) 18 (d) 72
20. 20.What is the product of the roots of x² − 14x + 48 = 0? (a) 47 (b) 14 (c) 48 (d) 49
21. 21.Using (a + b)² = a² + 2ab + b², what is 42²? (a) 1604 (b) 1444 (c) 1774 (d) 1764
22. 22.What are the roots of x² − 9x + 14 = 0? (a) 2 and 7 (b) 3 and 6 (c) 1 and 8 (d) 1 and 14
23. 23.Using (a + b)² = a² + 2ab + b², what is 34²? (a) 916 (b) 1166 (c) 676 (d) 1156
24. 24.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 202 (c) 400 (d) 598
25. 25.What is the product of the roots of x² − 13x + 36 = 0? (a) 37 (b) 36 (c) 13 (d) 35
26. 26.If a + b = 17 and ab = 70, what is (a − b)²? (a) 149 (b) 3 (c) 9 (d) 289
27. 27.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1575 (c) 1235 (d) 1225
28. 28.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 4225 (b) 4925 (c) 5625 (d) 5635
29. 29.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 256 (c) 193 (d) 382
30. 30.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 6)(x + 8) (c) (x + 1)(x + 48) (d) (x + 1)(x + 13)
31. 31.What are the roots of x² − 12x + 32 = 0? (a) 1 and 32 (b) 1 and 11 (c) 2 and 16 (d) 4 and 8
32. 32.If a + b = 13 and ab = 30, what is a² + b²? (a) 169 (b) 139 (c) 109 (d) 229
33. 33.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2475 (b) 2035 (c) 3025 (d) 2025
34. 34.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 21) (b) (x + 3)(x + 7) (c) (x + 1)(x + 9) (d) (x − 3)(x − 7)
35. 35.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1946 (c) 1936 (d) 1296
36. 36.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1296 (c) 1936 (d) 1584
37. 37.If a + b = 10 and ab = 24, what is a² + b²? (a) 76 (b) 148 (c) 52 (d) 100
38. 38.Using (a + b)² = a² + 2ab + b², what is 101²? (a) 9801 (b) 10001 (c) 10211 (d) 10201
39. 39.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1599 (b) 1521 (c) 1531 (d) 1681
40. 40.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 5 and 6 (c) 1 and 10 (d) 1 and 30

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