Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 281 · 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 103²? (a) 10609 (b) 9409 (c) 10619 (d) 10009
2. 2.What are the roots of x² − 17x + 72 = 0? (a) 1 and 16 (b) 2 and 36 (c) 8 and 9 (d) 1 and 72
3. 3.If a + b = 13 and ab = 42, what is a² + b²? (a) 85 (b) 127 (c) 169 (d) 253
4. 4.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
5. 5.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6241 (b) 6399 (c) 6561 (d) 6251
6. 6.If a + b = 19 and ab = 88, what is (a − b)²? (a) 185 (b) 361 (c) 3 (d) 9
7. 7.What are the roots of x² − 11x + 30 = 0? (a) 2 and 15 (b) 1 and 10 (c) 1 and 30 (d) 5 and 6
8. 8.Using (a + b)² = a² + 2ab + b², what is 21²? (a) 361 (b) 441 (c) 401 (d) 451
9. 9.Using (a − b)² = a² − 2ab + b², what is 68²? (a) 4624 (b) 4634 (c) 5184 (d) 4896
10. 10.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 9 (c) 3 (d) 29
11. 11.If a + b = 14 and ab = 40, what is (a − b)²? (a) 116 (b) 196 (c) 36 (d) 6
12. 12.If a + b = 15 and ab = 50, what is a² + b²? (a) 175 (b) 325 (c) 125 (d) 225
13. 13.What is the product of the roots of x² − 8x + 12 = 0? (a) 12 (b) 11 (c) 13 (d) 8
14. 14.What is the product of the roots of x² − 7x + 10 = 0? (a) 11 (b) 7 (c) 9 (d) 10
15. 15.If a + b = 10 and ab = 9, what is (a − b)²? (a) 82 (b) 100 (c) 8 (d) 64
16. 16.Using (a − b)² = a² − 2ab + b², what is 88²? (a) 8464 (b) 7754 (c) 8096 (d) 7744
17. 17.What is the sum of the roots of x² − 11x + 30 = 0? (a) 11 (b) 10 (c) 30 (d) 12
18. 18.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 324 (b) 396 (c) 484 (d) 334
19. 19.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 7406 (b) 8084 (c) 8836 (d) 7396
20. 20.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x + 1)(x + 20) (c) (x + 4)(x + 5) (d) (x − 4)(x − 5)
21. 21.If a + b = 15 and ab = 44, what is (a − b)²? (a) 49 (b) 137 (c) 225 (d) 7
22. 22.Which is the correct factorisation of x² + 6x + 8? (a) (x − 2)(x − 4) (b) (x + 1)(x + 8) (c) (x + 1)(x + 5) (d) (x + 2)(x + 4)
23. 23.What are the roots of x² − 6x + 8 = 0? (a) 2 and 4 (b) 2 and 5 (c) 1 and 5 (d) 1 and 8
24. 24.If a + b = 13 and ab = 42, what is (a − b)²? (a) 1 (b) 127 (c) 169 (d) 85
25. 25.If a + b = 19 and ab = 90, what is a² + b²? (a) 361 (b) 541 (c) 181 (d) 271
26. 26.What is the sum of the roots of x² − 13x + 42 = 0? (a) 42 (b) 13 (c) 14 (d) 12
27. 27.What is the sum of the roots of x² − 8x + 12 = 0? (a) 9 (b) 8 (c) 7 (d) 12
28. 28.Which is the correct factorisation of x² + 16x + 63? (a) (x + 1)(x + 63) (b) (x + 7)(x + 9) (c) (x − 7)(x − 9) (d) (x + 1)(x + 15)
29. 29.If a + b = 16 and ab = 60, what is (a − b)²? (a) 4 (b) 256 (c) 16 (d) 136
30. 30.Using (a − b)² = a² − 2ab + b², what is 35²? (a) 2025 (b) 1575 (c) 1235 (d) 1225
31. 31.What is the sum of the roots of x² − 15x + 56 = 0? (a) 14 (b) 15 (c) 56 (d) 16
32. 32.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 219 (c) 149 (d) 429
33. 33.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8099 (b) 7931 (c) 7921 (d) 8281
34. 34.Which is the correct factorisation of x² + 10x + 24? (a) (x + 4)(x + 6) (b) (x − 4)(x − 6) (c) (x + 1)(x + 24) (d) (x + 1)(x + 9)
35. 35.What is the sum of the roots of x² − 11x + 24 = 0? (a) 11 (b) 12 (c) 24 (d) 10
36. 36.Using (a − b)² = a² − 2ab + b², what is 38²? (a) 1596 (b) 1454 (c) 1444 (d) 1764
37. 37.What is the sum of the roots of x² − 6x + 8 = 0? (a) 6 (b) 8 (c) 5 (d) 7
38. 38.What is the sum of the roots of x² − 9x + 20 = 0? (a) 9 (b) 8 (c) 20 (d) 10
39. 39.If a + b = 16 and ab = 63, what is a² + b²? (a) 193 (b) 130 (c) 256 (d) 382
40. 40.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 1)(x + 21) (d) (x + 3)(x + 7)

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