Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 788 · 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 85²? (a) 8075 (b) 7235 (c) 9025 (d) 7225
2. 2.If a + b = 17 and ab = 60, what is a² + b²? (a) 229 (b) 289 (c) 169 (d) 409
3. 3.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 896 (d) 1024
4. 4.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 42) (c) (x + 1)(x + 12) (d) (x + 6)(x + 7)
5. 5.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4896 (b) 4634 (c) 4624 (d) 5184
6. 6.What is the sum of the roots of x² − 7x + 10 = 0? (a) 8 (b) 7 (c) 6 (d) 10
7. 7.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 1225 (b) 2025 (c) 1235 (d) 1575
8. 8.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 3616 (b) 4096 (c) 4106 (d) 3136
9. 9.Using (a − b)² = a² − 2ab + b², what is 39²? (a) 1521 (b) 1531 (c) 1681 (d) 1599
10. 10.Which is the correct factorisation of x² + 7x + 12? (a) (x + 3)(x + 4) (b) (x + 1)(x + 12) (c) (x + 1)(x + 6) (d) (x − 3)(x − 4)
11. 11.What is the sum of the roots of x² − 13x + 42 = 0? (a) 13 (b) 12 (c) 14 (d) 42
12. 12.What is the sum of the roots of x² − 12x + 35 = 0? (a) 12 (b) 35 (c) 13 (d) 11
13. 13.What are the roots of x² − 13x + 40 = 0? (a) 1 and 40 (b) 2 and 20 (c) 1 and 12 (d) 5 and 8
14. 14.If a + b = 18 and ab = 77, what is a² + b²? (a) 247 (b) 170 (c) 324 (d) 478
15. 15.What are the roots of x² − 5x + 6 = 0? (a) 2 and 3 (b) 1 and 4 (c) 1 and 6 (d) 2 and 4
16. 16.Using (a − b)² = a² − 2ab + b², what is 15²? (a) 625 (b) 225 (c) 235 (d) 375
17. 17.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 72 (c) 73 (d) 17
18. 18.What are the roots of x² − 14x + 48 = 0? (a) 2 and 24 (b) 1 and 48 (c) 1 and 13 (d) 6 and 8
19. 19.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 7)(x + 8)
20. 20.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 529 (c) 539 (d) 409
21. 21.If a + b = 11 and ab = 28, what is a² + b²? (a) 177 (b) 93 (c) 65 (d) 121
22. 22.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 13 (c) 12 (d) 14
23. 23.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 16 (d) 15
24. 24.Which is the correct factorisation of x² + 15x + 54? (a) (x + 6)(x + 9) (b) (x − 6)(x − 9) (c) (x + 1)(x + 14) (d) (x + 1)(x + 54)
25. 25.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 11 (c) 18 (d) 10
26. 26.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 401 (c) 451 (d) 361
27. 27.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 400 (c) 2 (d) 202
28. 28.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1616 (c) 1946 (d) 1936
29. 29.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 193 (c) 157 (d) 145
30. 30.If a + b = 7 and ab = 10, what is a² + b²? (a) 39 (b) 49 (c) 29 (d) 69
31. 31.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2126 (c) 2484 (d) 2116
32. 32.If a + b = 17 and ab = 66, what is a² + b²? (a) 421 (b) 157 (c) 289 (d) 223
33. 33.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)
34. 34.Using (a − b)² = a² − 2ab + b², what is 37²? (a) 1849 (b) 1591 (c) 1369 (d) 1379
35. 35.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 15) (c) (x + 7)(x + 9) (d) (x + 1)(x + 63)
36. 36.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)
37. 37.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 9035 (c) 9975 (d) 11025
38. 38.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 2 (c) 16 (d) 10
39. 39.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 144 (c) 74 (d) 214
40. 40.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 8 (c) 10 (d) 20

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