Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 141 · 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.What are the roots of x² − 14x + 48 = 0? (a) 6 and 8 (b) 1 and 48 (c) 2 and 24 (d) 1 and 13
2. 2.If a + b = 9 and ab = 18, what is a² + b²? (a) 81 (b) 45 (c) 117 (d) 63
3. 3.Which is the correct factorisation of x² + 12x + 27? (a) (x + 3)(x + 9) (b) (x + 1)(x + 11) (c) (x − 3)(x − 9) (d) (x + 1)(x + 27)
4. 4.Using (a − b)² = a² − 2ab + b², what is 55²? (a) 3025 (b) 3575 (c) 3035 (d) 4225
5. 5.Which is the correct factorisation of x² + 9x + 18? (a) (x − 3)(x − 6) (b) (x + 3)(x + 6) (c) (x + 1)(x + 8) (d) (x + 1)(x + 18)
6. 6.If a + b = 19 and ab = 88, what is a² + b²? (a) 185 (b) 361 (c) 537 (d) 273
7. 7.Which is the correct factorisation of x² + 11x + 28? (a) (x − 4)(x − 7) (b) (x + 1)(x + 28) (c) (x + 4)(x + 7) (d) (x + 1)(x + 10)
8. 8.Using (a + b)² = a² + 2ab + b², what is 85²? (a) 5625 (b) 7225 (c) 6425 (d) 7235
9. 9.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
10. 10.What are the roots of x² − 12x + 32 = 0? (a) 4 and 8 (b) 1 and 11 (c) 2 and 16 (d) 1 and 32
11. 11.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 10 (c) 9 (d) 14
12. 12.Using (a − b)² = a² − 2ab + b², what is 86²? (a) 8084 (b) 8836 (c) 7406 (d) 7396
13. 13.If a + b = 20 and ab = 96, what is a² + b²? (a) 400 (b) 208 (c) 592 (d) 304
14. 14.Which is the correct factorisation of x² + 17x + 72? (a) (x + 1)(x + 72) (b) (x + 1)(x + 16) (c) (x + 8)(x + 9) (d) (x − 8)(x − 9)
15. 15.If a + b = 21 and ab = 108, what is (a − b)²? (a) 441 (b) 9 (c) 225 (d) 3
16. 16.If a + b = 19 and ab = 84, what is (a − b)²? (a) 361 (b) 193 (c) 5 (d) 25
17. 17.If a + b = 16 and ab = 60, what is a² + b²? (a) 136 (b) 376 (c) 196 (d) 256
18. 18.What is the product of the roots of x² − 14x + 48 = 0? (a) 49 (b) 48 (c) 47 (d) 14
19. 19.If a + b = 14 and ab = 40, what is a² + b²? (a) 196 (b) 156 (c) 276 (d) 116
20. 20.Which is the correct factorisation of x² + 11x + 18? (a) (x + 1)(x + 10) (b) (x − 2)(x − 9) (c) (x + 2)(x + 9) (d) (x + 1)(x + 18)
21. 21.If a + b = 17 and ab = 72, what is (a − b)²? (a) 145 (b) 217 (c) 289 (d) 1
22. 22.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
23. 23.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x − 2)(x − 4) (c) (x + 1)(x + 5) (d) (x + 1)(x + 8)
24. 24.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 9 (c) 6 (d) 7
25. 25.Which is the correct factorisation of x² + 8x + 15? (a) (x + 3)(x + 5) (b) (x − 3)(x − 5) (c) (x + 1)(x + 7) (d) (x + 1)(x + 15)
26. 26.Which is the correct factorisation of x² + 7x + 10? (a) (x − 2)(x − 5) (b) (x + 1)(x + 6) (c) (x + 1)(x + 10) (d) (x + 2)(x + 5)
27. 27.If a + b = 17 and ab = 72, what is a² + b²? (a) 145 (b) 289 (c) 433 (d) 217
28. 28.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 6) (b) (x + 3)(x + 4) (c) (x − 3)(x − 4) (d) (x + 1)(x + 12)
29. 29.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 396 (b) 334 (c) 484 (d) 324
30. 30.Using (a + b)² = a² + 2ab + b², what is 93²? (a) 8659 (b) 8649 (c) 7569 (d) 8109
31. 31.If a + b = 12 and ab = 35, what is a² + b²? (a) 144 (b) 214 (c) 109 (d) 74
32. 32.If a + b = 4 and ab = 3, what is a² + b²? (a) 16 (b) 10 (c) 13 (d) 22
33. 33.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 17 (d) 72
34. 34.If a + b = 16 and ab = 48, what is (a − b)²? (a) 8 (b) 64 (c) 160 (d) 256
35. 35.If a + b = 8 and ab = 7, what is (a − b)²? (a) 6 (b) 36 (c) 64 (d) 50
36. 36.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 399 (d) 441
37. 37.If a + b = 11 and ab = 24, what is a² + b²? (a) 73 (b) 97 (c) 169 (d) 121
38. 38.If a + b = 19 and ab = 84, what is a² + b²? (a) 529 (b) 277 (c) 193 (d) 361
39. 39.Using (a + b)² = a² + 2ab + b², what is 54²? (a) 2926 (b) 2116 (c) 2516 (d) 2916
40. 40.What is the sum of the roots of x² − 14x + 45 = 0? (a) 13 (b) 45 (c) 15 (d) 14

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