Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 671 · 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 = 10 and ab = 16, what is (a − b)²? (a) 6 (b) 68 (c) 100 (d) 36
2. 2.What is the sum of the roots of x² − 12x + 27 = 0? (a) 11 (b) 27 (c) 12 (d) 13
3. 3.Which is the correct factorisation of x² + 13x + 36? (a) (x − 4)(x − 9) (b) (x + 4)(x + 9) (c) (x + 1)(x + 12) (d) (x + 1)(x + 36)
4. 4.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9991 (b) 10609 (c) 9419 (d) 9409
5. 5.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6251 (c) 6561 (d) 6399
6. 6.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 180 (c) 252 (d) 468
7. 7.What is the sum of the roots of x² − 11x + 24 = 0? (a) 10 (b) 24 (c) 11 (d) 12
8. 8.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 3)(x + 9) (c) (x + 1)(x + 11) (d) (x + 1)(x + 27)
9. 9.If a + b = 20 and ab = 96, what is a² + b²? (a) 592 (b) 400 (c) 208 (d) 304
10. 10.If a + b = 17 and ab = 60, what is a² + b²? (a) 409 (b) 229 (c) 289 (d) 169
11. 11.What are the roots of x² − 12x + 27 = 0? (a) 1 and 27 (b) 3 and 9 (c) 1 and 11 (d) 2 and 10
12. 12.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 5625 (b) 4875 (c) 4235 (d) 4225
13. 13.What is the sum of the roots of x² − 9x + 18 = 0? (a) 8 (b) 18 (c) 10 (d) 9
14. 14.Using (a − b)² = a² − 2ab + b², what is 48²? (a) 2314 (b) 2704 (c) 2304 (d) 2496
15. 15.Using (a − b)² = a² − 2ab + b², what is 75²? (a) 7225 (b) 6375 (c) 5625 (d) 5635
16. 16.What are the roots of x² − 13x + 42 = 0? (a) 1 and 42 (b) 6 and 7 (c) 2 and 21 (d) 1 and 12
17. 17.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 4884 (b) 4356 (c) 4366 (d) 5476
18. 18.Which is the correct factorisation of x² + 7x + 10? (a) (x + 1)(x + 10) (b) (x − 2)(x − 5) (c) (x + 2)(x + 5) (d) (x + 1)(x + 6)
19. 19.If a + b = 7 and ab = 10, what is a² + b²? (a) 49 (b) 29 (c) 69 (d) 39
20. 20.Using (a + b)² = a² + 2ab + b², what is 82²? (a) 6734 (b) 6084 (c) 6404 (d) 6724
21. 21.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 16) (b) (x + 8)(x + 9) (c) (x − 8)(x − 9) (d) (x + 1)(x + 72)
22. 22.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
23. 23.What are the roots of x² − 11x + 30 = 0? (a) 1 and 10 (b) 2 and 15 (c) 5 and 6 (d) 1 and 30
24. 24.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1296 (b) 1616 (c) 1936 (d) 1946
25. 25.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1764 (b) 1596 (c) 1444 (d) 1454
26. 26.What is the sum of the roots of x² − 15x + 56 = 0? (a) 15 (b) 56 (c) 16 (d) 14
27. 27.If a + b = 15 and ab = 36, what is a² + b²? (a) 225 (b) 189 (c) 153 (d) 297
28. 28.What is the sum of the roots of x² − 11x + 28 = 0? (a) 10 (b) 12 (c) 11 (d) 28
29. 29.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)
30. 30.What is the product of the roots of x² − 11x + 18 = 0? (a) 19 (b) 18 (c) 11 (d) 17
31. 31.Which is the correct factorisation of x² + 5x + 6? (a) (x + 1)(x + 6) (b) (x − 2)(x − 3) (c) (x + 1)(x + 4) (d) (x + 2)(x + 3)
32. 32.If a + b = 13 and ab = 36, what is (a − b)²? (a) 25 (b) 97 (c) 5 (d) 169
33. 33.If a + b = 12 and ab = 27, what is a² + b²? (a) 90 (b) 198 (c) 117 (d) 144
34. 34.If a + b = 20 and ab = 99, what is (a − b)²? (a) 2 (b) 4 (c) 400 (d) 202
35. 35.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 784 (c) 1024 (d) 896
36. 36.What is the product of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 71 (d) 73
37. 37.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2035 (c) 2025 (d) 1225
38. 38.If a + b = 10 and ab = 21, what is a² + b²? (a) 142 (b) 58 (c) 79 (d) 100
39. 39.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 7235 (b) 5625 (c) 7225 (d) 6425
40. 40.If a + b = 15 and ab = 44, what is a² + b²? (a) 225 (b) 181 (c) 313 (d) 137

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