Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 893 · 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 = 11 and ab = 30, what is a² + b²? (a) 91 (b) 61 (c) 181 (d) 121
2. 2.What are the roots of x² − 6x + 8 = 0? (a) 2 and 5 (b) 1 and 5 (c) 1 and 8 (d) 2 and 4
3. 3.If a + b = 18 and ab = 80, what is a² + b²? (a) 324 (b) 164 (c) 484 (d) 244
4. 4.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 13 (c) 10 (d) 22
5. 5.What is the product of the roots of x² − 8x + 15 = 0? (a) 15 (b) 8 (c) 14 (d) 16
6. 6.If a + b = 20 and ab = 99, what is (a − b)²? (a) 4 (b) 202 (c) 2 (d) 400
7. 7.If a + b = 16 and ab = 63, what is (a − b)²? (a) 130 (b) 2 (c) 4 (d) 256
8. 8.Using (a − b)² = a² − 2ab + b², what is 79²? (a) 6251 (b) 6399 (c) 6241 (d) 6561
9. 9.If a + b = 13 and ab = 42, what is (a − b)²? (a) 85 (b) 1 (c) 169 (d) 127
10. 10.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)
11. 11.Which is the correct factorisation of x² + 10x + 21? (a) (x + 1)(x + 9) (b) (x − 3)(x − 7) (c) (x + 3)(x + 7) (d) (x + 1)(x + 21)
12. 12.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 576 (b) 256 (c) 266 (d) 384
13. 13.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 1)(x + 16) (d) (x + 8)(x + 9)
14. 14.If a + b = 8 and ab = 15, what is (a − b)²? (a) 2 (b) 64 (c) 34 (d) 4
15. 15.If a + b = 12 and ab = 11, what is (a − b)²? (a) 10 (b) 144 (c) 122 (d) 100
16. 16.Which is the correct factorisation of x² + 15x + 56? (a) (x − 7)(x − 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x + 7)(x + 8)
17. 17.If a + b = 14 and ab = 33, what is (a − b)²? (a) 64 (b) 130 (c) 196 (d) 8
18. 18.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 10609 (b) 9991 (c) 9419 (d) 9409
19. 19.Which is the correct factorisation of x² + 11x + 28? (a) (x + 4)(x + 7) (b) (x − 4)(x − 7) (c) (x + 1)(x + 28) (d) (x + 1)(x + 10)
20. 20.What is the sum of the roots of x² − 17x + 72 = 0? (a) 16 (b) 18 (c) 17 (d) 72
21. 21.Which is the correct factorisation of x² + 14x + 48? (a) (x + 6)(x + 8) (b) (x + 1)(x + 13) (c) (x − 6)(x − 8) (d) (x + 1)(x + 48)
22. 22.Which is the correct factorisation of x² + 13x + 42? (a) (x − 6)(x − 7) (b) (x + 1)(x + 12) (c) (x + 6)(x + 7) (d) (x + 1)(x + 42)
23. 23.If a + b = 8 and ab = 7, what is (a − b)²? (a) 36 (b) 6 (c) 64 (d) 50
24. 24.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 4356 (c) 5486 (d) 5476
25. 25.If a + b = 9 and ab = 20, what is (a − b)²? (a) 61 (b) 41 (c) 81 (d) 1
26. 26.If a + b = 16 and ab = 63, what is a² + b²? (a) 130 (b) 193 (c) 256 (d) 382
27. 27.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 289 (b) 529 (c) 391 (d) 299
28. 28.What is the sum of the roots of x² − 11x + 30 = 0? (a) 10 (b) 30 (c) 12 (d) 11
29. 29.If a + b = 12 and ab = 35, what is a² + b²? (a) 109 (b) 214 (c) 74 (d) 144
30. 30.Using (a + b)² = a² + 2ab + b², what is 72²? (a) 5194 (b) 4904 (c) 4624 (d) 5184
31. 31.Using (a + b)² = a² + 2ab + b², what is 81²? (a) 6571 (b) 6401 (c) 6561 (d) 6241
32. 32.Using (a − b)² = a² − 2ab + b², what is 36²? (a) 1936 (b) 1584 (c) 1306 (d) 1296
33. 33.What is the sum of the roots of x² − 13x + 36 = 0? (a) 36 (b) 14 (c) 13 (d) 12
34. 34.What are the roots of x² − 7x + 12 = 0? (a) 3 and 4 (b) 1 and 12 (c) 2 and 6 (d) 1 and 6
35. 35.If a + b = 20 and ab = 99, what is a² + b²? (a) 301 (b) 598 (c) 400 (d) 202
36. 36.If a + b = 13 and ab = 12, what is a² + b²? (a) 169 (b) 157 (c) 145 (d) 193
37. 37.If a + b = 19 and ab = 90, what is (a − b)²? (a) 361 (b) 181 (c) 271 (d) 1
38. 38.Using (a − b)² = a² − 2ab + b², what is 66²? (a) 5476 (b) 4884 (c) 4356 (d) 4366
39. 39.What is the product of the roots of x² − 11x + 24 = 0? (a) 23 (b) 25 (c) 11 (d) 24
40. 40.Using (a − b)² = a² − 2ab + b², what is 28²? (a) 794 (b) 1024 (c) 896 (d) 784

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