Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 317 · 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 75²? (a) 5635 (b) 5625 (c) 4225 (d) 4925
2. 2.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 9 (d) 225
3. 3.If a + b = 5 and ab = 4, what is a² + b²? (a) 17 (b) 33 (c) 21 (d) 25
4. 4.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x − 7)(x − 8) (c) (x + 1)(x + 56) (d) (x + 1)(x + 14)
5. 5.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 1 and 72 (c) 8 and 9 (d) 2 and 36
6. 6.What is the sum of the roots of x² − 7x + 10 = 0? (a) 10 (b) 6 (c) 7 (d) 8
7. 7.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6561 (c) 6251 (d) 6399
8. 8.Using (a + b)² = a² + 2ab + b², what is 23²? (a) 289 (b) 529 (c) 539 (d) 409
9. 9.What is the sum of the roots of x² − 16x + 63 = 0? (a) 15 (b) 17 (c) 63 (d) 16
10. 10.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 598 (c) 400 (d) 202
11. 11.If a + b = 15 and ab = 54, what is (a − b)²? (a) 9 (b) 3 (c) 225 (d) 117
12. 12.If a + b = 8 and ab = 7, what is (a − b)²? (a) 64 (b) 50 (c) 6 (d) 36
13. 13.If a + b = 16 and ab = 60, what is a² + b²? (a) 376 (b) 196 (c) 256 (d) 136
14. 14.If a + b = 14 and ab = 48, what is (a − b)²? (a) 100 (b) 196 (c) 2 (d) 4
15. 15.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7225 (b) 7235 (c) 8075 (d) 9025
16. 16.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 7) (b) (x + 1)(x + 15) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
17. 17.What is the sum of the roots of x² − 11x + 30 = 0? (a) 30 (b) 12 (c) 10 (d) 11
18. 18.If a + b = 19 and ab = 90, what is (a − b)²? (a) 181 (b) 1 (c) 361 (d) 271
19. 19.Which is the correct factorisation of x² + 11x + 18? (a) (x − 2)(x − 9) (b) (x + 1)(x + 10) (c) (x + 1)(x + 18) (d) (x + 2)(x + 9)
20. 20.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)
21. 21.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 81 (c) 121 (d) 41
22. 22.What are the roots of x² − 16x + 63 = 0? (a) 1 and 15 (b) 7 and 9 (c) 3 and 21 (d) 1 and 63
23. 23.Using (a + b)² = a² + 2ab + b², what is 64²? (a) 4106 (b) 3136 (c) 3616 (d) 4096
24. 24.Which is the correct factorisation of x² + 13x + 40? (a) (x − 5)(x − 8) (b) (x + 1)(x + 12) (c) (x + 1)(x + 40) (d) (x + 5)(x + 8)
25. 25.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1946 (b) 1296 (c) 1616 (d) 1936
26. 26.What is the product of the roots of x² − 11x + 30 = 0? (a) 11 (b) 29 (c) 31 (d) 30
27. 27.What are the roots of x² − 12x + 35 = 0? (a) 2 and 10 (b) 5 and 7 (c) 1 and 35 (d) 1 and 11
28. 28.Using (a + b)² = a² + 2ab + b², what is 62²? (a) 3604 (b) 3364 (c) 3844 (d) 3854
29. 29.If a + b = 18 and ab = 72, what is (a − b)²? (a) 324 (b) 6 (c) 36 (d) 180
30. 30.If a + b = 10 and ab = 9, what is (a − b)²? (a) 64 (b) 8 (c) 82 (d) 100
31. 31.What is the sum of the roots of x² − 12x + 32 = 0? (a) 11 (b) 32 (c) 12 (d) 13
32. 32.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1306 (b) 1936 (c) 1296 (d) 1584
33. 33.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 25 (c) 97 (d) 5
34. 34.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x + 1)(x + 16) (d) (x − 2)(x − 8)
35. 35.What is the product of the roots of x² − 15x + 56 = 0? (a) 57 (b) 55 (c) 15 (d) 56
36. 36.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 4) (b) (x − 2)(x − 3) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
37. 37.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 6375 (c) 5635 (d) 5625
38. 38.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8649 (b) 7569 (c) 8659 (d) 8109
39. 39.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1521 (b) 1601 (c) 1681 (d) 1691
40. 40.What are the roots of x² − 15x + 54 = 0? (a) 1 and 54 (b) 6 and 9 (c) 1 and 14 (d) 2 and 27

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