Question

Вычислите (1+i)(2+i)+ 5/(1+2i)

Answer 1

$2 + i + 2i + {i}^{2} \: + \frac{5}{1 + 2i} \: = 2 + i$

Answer 2

To calculate the expression, let's first perform the operations step by step:

1. (1 + i)(2 + i): Expand the product using the distributive property: (1 + i)(2 + i) = 1 * 2 + 1 * i + i * 2 + i * i (1 + i)(2 + i) = 2 + i + 2i + i^2

   Recall that i^2 is equal to -1: (1 + i)(2 + i) = 2 + 3i - 1 (1 + i)(2 + i) = 1 + 3i

2. 5/(1 + 2i): To get rid of the complex denominator, multiply the numerator and denominator by the conjugate of the denominator: The conjugate of (1 + 2i) is (1 - 2i). So, (5 / (1 + 2i)) * ((1 - 2i) / (1 - 2i)) = (5 * (1 - 2i)) / ((1 + 2i) * (1 - 2i))

   Now, calculate the denominator: (1 + 2i) * (1 - 2i) = 1 - 2i + 2i - 4i^2 Recall that i^2 is equal to -1: (1 + 2i) * (1 - 2i) = 1 - 4(-1) (1 + 2i) * (1 - 2i) = 1 + 4 (1 + 2i) * (1 - 2i) = 5

   Now, the expression becomes: (5 * (1 - 2i)) / 5 = (5 - 10i) / 5

   Finally, simplify: (5 - 10i) / 5 = 5/5 - (10i)/5 (5 - 10i) / 5 = 1 - 2i

Now, add the results from step 1 and step 2:

(1 + i)(2 + i) + 5/(1 + 2i) = (1 + 3i) + (1 - 2i)

Combine like terms:

(1 + 3i) + (1 - 2i) = 1 + 1 + 3i - 2i

Simplify:

1 + 1 = 2

The final result is:

(1 + i)(2 + i) + 5/(1 + 2i) = 2 + i