Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 569 · 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 = 12 and ab = 27, what is (a − b)²? (a) 90 (b) 144 (c) 36 (d) 6
2. 2.What is the sum of the roots of x² − 12x + 35 = 0? (a) 13 (b) 12 (c) 35 (d) 11
3. 3.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 164 (c) 244 (d) 324
4. 4.What is the sum of the roots of x² − 16x + 63 = 0? (a) 16 (b) 17 (c) 63 (d) 15
5. 5.Using (a − b)² = a² − 2ab + b², what is 26²? (a) 1156 (b) 676 (c) 884 (d) 686
6. 6.Using (a + b)² = a² + 2ab + b², what is 35²? (a) 925 (b) 1235 (c) 625 (d) 1225
7. 7.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 4904 (b) 5194 (c) 4624 (d) 5184
8. 8.What are the roots of x² − 17x + 72 = 0? (a) 1 and 72 (b) 2 and 36 (c) 1 and 16 (d) 8 and 9
9. 9.What is the sum of the roots of x² − 6x + 8 = 0? (a) 5 (b) 8 (c) 7 (d) 6
10. 10.If a + b = 20 and ab = 96, what is (a − b)²? (a) 208 (b) 4 (c) 16 (d) 400
11. 11.What is the product of the roots of x² − 11x + 18 = 0? (a) 17 (b) 19 (c) 18 (d) 11
12. 12.If a + b = 15 and ab = 36, what is a² + b²? (a) 297 (b) 189 (c) 153 (d) 225
13. 13.Using (a − b)² = a² − 2ab + b², what is 45²? (a) 3025 (b) 2035 (c) 2025 (d) 2475
14. 14.What are the roots of x² − 14x + 48 = 0? (a) 1 and 13 (b) 6 and 8 (c) 2 and 24 (d) 1 and 48
15. 15.Using (a − b)² = a² − 2ab + b², what is 85²? (a) 7235 (b) 9025 (c) 8075 (d) 7225
16. 16.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 441 (b) 451 (c) 401 (d) 361
17. 17.What is the product of the roots of x² − 17x + 72 = 0? (a) 71 (b) 17 (c) 73 (d) 72
18. 18.Using (a − b)² = a² − 2ab + b², what is 69²? (a) 4761 (b) 4899 (c) 4771 (d) 5041
19. 19.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 18) (b) (x + 1)(x + 10) (c) (x − 2)(x − 9) (d) (x + 2)(x + 9)
20. 20.If a + b = 18 and ab = 77, what is (a − b)²? (a) 170 (b) 16 (c) 324 (d) 4
21. 21.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 271 (c) 181 (d) 541
22. 22.If a + b = 10 and ab = 9, what is a² + b²? (a) 100 (b) 118 (c) 91 (d) 82
23. 23.Which is the correct factorisation of x² + 12x + 35? (a) (x − 5)(x − 7) (b) (x + 1)(x + 11) (c) (x + 1)(x + 35) (d) (x + 5)(x + 7)
24. 24.Which is the correct factorisation of x² + 7x + 10? (a) (x + 2)(x + 5) (b) (x + 1)(x + 10) (c) (x − 2)(x − 5) (d) (x + 1)(x + 6)
25. 25.What is the sum of the roots of x² − 17x + 72 = 0? (a) 72 (b) 16 (c) 17 (d) 18
26. 26.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7066 (c) 7056 (d) 5776
27. 27.What is the product of the roots of x² − 15x + 56 = 0? (a) 56 (b) 15 (c) 55 (d) 57
28. 28.Which is the correct factorisation of x² + 12x + 27? (a) (x − 3)(x − 9) (b) (x + 1)(x + 27) (c) (x + 1)(x + 11) (d) (x + 3)(x + 9)
29. 29.Which is the correct factorisation of x² + 13x + 36? (a) (x + 4)(x + 9) (b) (x + 1)(x + 36) (c) (x − 4)(x − 9) (d) (x + 1)(x + 12)
30. 30.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 15) (b) (x + 1)(x + 63) (c) (x − 7)(x − 9) (d) (x + 7)(x + 9)
31. 31.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
32. 32.If a + b = 16 and ab = 63, what is a² + b²? (a) 256 (b) 193 (c) 130 (d) 382
33. 33.Using (a + b)² = a² + 2ab + b², what is 91²? (a) 8101 (b) 7921 (c) 8291 (d) 8281
34. 34.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 30) (b) (x + 1)(x + 10) (c) (x + 5)(x + 6) (d) (x − 5)(x − 6)
35. 35.If a + b = 13 and ab = 42, what is (a − b)²? (a) 169 (b) 85 (c) 1 (d) 127
36. 36.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 3 (c) 9 (d) 225
37. 37.Using (a − b)² = a² − 2ab + b², what is 78²? (a) 6396 (b) 6094 (c) 6084 (d) 6724
38. 38.Which is the correct factorisation of x² + 14x + 45? (a) (x − 5)(x − 9) (b) (x + 1)(x + 45) (c) (x + 5)(x + 9) (d) (x + 1)(x + 13)
39. 39.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 334 (c) 324 (d) 396
40. 40.Using (a + b)² = a² + 2ab + b², what is 61²? (a) 3601 (b) 3481 (c) 3731 (d) 3721

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