Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 493 · 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 = 14 and ab = 33, what is a² + b²? (a) 163 (b) 130 (c) 196 (d) 262
2. 2.Using (a + b)² = a² + 2ab + b², what is 45²? (a) 1625 (b) 2025 (c) 1225 (d) 2035
3. 3.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 361 (c) 3 (d) 9
4. 4.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)
5. 5.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 10816 (b) 9216 (c) 10016 (d) 10826
6. 6.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x − 5)(x − 6) (c) (x + 1)(x + 30) (d) (x + 5)(x + 6)
7. 7.What is the sum of the roots of x² − 9x + 18 = 0? (a) 9 (b) 8 (c) 10 (d) 18
8. 8.If a + b = 9 and ab = 8, what is (a − b)²? (a) 81 (b) 49 (c) 7 (d) 65
9. 9.Using (a − b)² = a² − 2ab + b², what is 27²? (a) 1089 (b) 891 (c) 729 (d) 739
10. 10.What is the product of the roots of x² − 9x + 14 = 0? (a) 15 (b) 14 (c) 9 (d) 13
11. 11.If a + b = 14 and ab = 24, what is a² + b²? (a) 244 (b) 196 (c) 148 (d) 172
12. 12.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1225 (c) 1235 (d) 625
13. 13.If a + b = 5 and ab = 6, what is (a − b)²? (a) 13 (b) 1 (c) 19 (d) 25
14. 14.If a + b = 18 and ab = 77, what is (a − b)²? (a) 4 (b) 324 (c) 170 (d) 16
15. 15.What is the product of the roots of x² − 13x + 40 = 0? (a) 41 (b) 39 (c) 40 (d) 13
16. 16.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 5776 (c) 7066 (d) 6416
17. 17.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 5)(x + 7) (c) (x + 1)(x + 35) (d) (x + 1)(x + 11)
18. 18.Which is the correct factorisation of x² + 10x + 16? (a) (x + 1)(x + 9) (b) (x + 2)(x + 8) (c) (x − 2)(x − 8) (d) (x + 1)(x + 16)
19. 19.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2499 (b) 2411 (c) 2601 (d) 2401
20. 20.If a + b = 19 and ab = 84, what is (a − b)²? (a) 25 (b) 5 (c) 193 (d) 361
21. 21.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 12 (c) 13 (d) 11
22. 22.If a + b = 15 and ab = 54, what is (a − b)²? (a) 225 (b) 9 (c) 3 (d) 117
23. 23.What is the sum of the roots of x² − 13x + 40 = 0? (a) 40 (b) 14 (c) 13 (d) 12
24. 24.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 144 (c) 124 (d) 104
25. 25.Using (a + b)² = a² + 2ab + b², what is 25²? (a) 225 (b) 635 (c) 625 (d) 425
26. 26.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 17 (c) 18 (d) 16
27. 27.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 851 (b) 961 (c) 899 (d) 841
28. 28.What is the sum of the roots of x² − 15x + 54 = 0? (a) 14 (b) 54 (c) 15 (d) 16
29. 29.What are the roots of x² − 11x + 28 = 0? (a) 1 and 28 (b) 1 and 10 (c) 2 and 14 (d) 4 and 7
30. 30.If a + b = 9 and ab = 18, what is (a − b)²? (a) 45 (b) 81 (c) 9 (d) 3
31. 31.If a + b = 9 and ab = 20, what is (a − b)²? (a) 81 (b) 61 (c) 41 (d) 1
32. 32.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 289 (c) 9 (d) 149
33. 33.What is the sum of the roots of x² − 12x + 35 = 0? (a) 11 (b) 35 (c) 13 (d) 12
34. 34.If a + b = 19 and ab = 90, what is a² + b²? (a) 541 (b) 361 (c) 271 (d) 181
35. 35.What is the product of the roots of x² − 11x + 24 = 0? (a) 25 (b) 24 (c) 11 (d) 23
36. 36.What is the sum of the roots of x² − 5x + 6 = 0? (a) 7 (b) 6 (c) 4 (d) 5
37. 37.If a + b = 13 and ab = 22, what is (a − b)²? (a) 169 (b) 125 (c) 81 (d) 9
38. 38.If a + b = 10 and ab = 21, what is a² + b²? (a) 100 (b) 79 (c) 142 (d) 58
39. 39.Using (a + b)² = a² + 2ab + b², what is 33²? (a) 909 (b) 1099 (c) 1089 (d) 729
40. 40.If a + b = 17 and ab = 60, what is a² + b²? (a) 169 (b) 229 (c) 289 (d) 409

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