Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 36 · 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 = 21 and ab = 108, what is (a − b)²? (a) 225 (b) 441 (c) 9 (d) 3
2. 2.If a + b = 7 and ab = 6, what is (a − b)²? (a) 5 (b) 37 (c) 49 (d) 25
3. 3.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 9 (c) 225 (d) 153
4. 4.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 4 (c) 16 (d) 170
5. 5.Which is the correct factorisation of x² + 13x + 40? (a) (x + 1)(x + 12) (b) (x + 1)(x + 40) (c) (x + 5)(x + 8) (d) (x − 5)(x − 8)
6. 6.What is the sum of the roots of x² − 16x + 63 = 0? (a) 17 (b) 63 (c) 16 (d) 15
7. 7.If a + b = 19 and ab = 88, what is (a − b)²? (a) 3 (b) 361 (c) 185 (d) 9
8. 8.If a + b = 17 and ab = 60, what is (a − b)²? (a) 49 (b) 289 (c) 169 (d) 7
9. 9.Using (a + b)² = a² + 2ab + b², what is 103²? (a) 10009 (b) 10609 (c) 10619 (d) 9409
10. 10.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x − 5)(x − 6) (c) (x + 1)(x + 10) (d) (x + 5)(x + 6)
11. 11.What is the product of the roots of x² − 10x + 21 = 0? (a) 21 (b) 10 (c) 20 (d) 22
12. 12.What is the product of the roots of x² − 7x + 12 = 0? (a) 11 (b) 13 (c) 12 (d) 7
13. 13.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 6) (c) (x − 2)(x − 5) (d) (x + 1)(x + 10)
14. 14.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 14) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
15. 15.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 193 (c) 157 (d) 145
16. 16.If a + b = 5 and ab = 4, what is (a − b)²? (a) 17 (b) 25 (c) 3 (d) 9
17. 17.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6724 (b) 6084 (c) 6396 (d) 6094
18. 18.Using (a − b)² = a² − 2ab + b², what is 25²? (a) 635 (b) 1225 (c) 875 (d) 625
19. 19.What is the product of the roots of x² − 12x + 32 = 0? (a) 31 (b) 32 (c) 12 (d) 33
20. 20.If a + b = 11 and ab = 10, what is a² + b²? (a) 141 (b) 121 (c) 111 (d) 101
21. 21.Using (a + b)² = a² + 2ab + b², what is 41²? (a) 1691 (b) 1521 (c) 1601 (d) 1681
22. 22.If a + b = 17 and ab = 70, what is a² + b²? (a) 429 (b) 289 (c) 219 (d) 149
23. 23.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 851 (c) 961 (d) 841
24. 24.If a + b = 20 and ab = 99, what is a² + b²? (a) 400 (b) 598 (c) 301 (d) 202
25. 25.Using (a + b)² = a² + 2ab + b², what is 75²? (a) 5625 (b) 4225 (c) 5635 (d) 4925
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.If a + b = 15 and ab = 56, what is a² + b²? (a) 225 (b) 169 (c) 337 (d) 113
28. 28.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 896 (d) 784
29. 29.What is the product of the roots of x² − 17x + 72 = 0? (a) 73 (b) 71 (c) 72 (d) 17
30. 30.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2116 (c) 2916 (d) 2516
31. 31.If a + b = 14 and ab = 48, what is (a − b)²? (a) 4 (b) 100 (c) 196 (d) 2
32. 32.What are the roots of x² − 11x + 28 = 0? (a) 4 and 7 (b) 1 and 28 (c) 1 and 10 (d) 2 and 14
33. 33.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 5929 (b) 6899 (c) 6889 (d) 6409
34. 34.If a + b = 11 and ab = 24, what is (a − b)²? (a) 5 (b) 121 (c) 73 (d) 25
35. 35.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 5184 (c) 4624 (d) 4904
36. 36.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10016 (c) 10816 (d) 10826
37. 37.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 2 (c) 4 (d) 202
38. 38.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2504 (c) 2714 (d) 2304
39. 39.What is the product of the roots of x² − 13x + 36 = 0? (a) 35 (b) 36 (c) 37 (d) 13
40. 40.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 2035 (b) 2475 (c) 2025 (d) 3025

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