Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$\frac{2 i}{2+i}+\frac{5}{2-i}$$

Step-by-Step Solution

Answer

The result in standard form is \(3 + \frac{3}{5}i\).

Step 1: Get the Complex Conjugates

First, get the complex conjugates of the denominators to simplify the fractions: the conjugate of \(2 + i\) is \(2 - i\), and the conjugate of \(2 - i\) is \(2 + i\).

Step 2: Multiply numerators and denominators by the conjugate

Multiply the numerators and the denominators of each of the fractions by their respective complex conjugate. \[\frac{2i}{2+i} * \frac{2-i}{2-i} + \frac{5}{2-i} * \frac{2+i}{2+i}\]

Step 3: Distribute and Simplify

Perform multiplication and simplify the fractions. \[\frac{2i(2-i)}{(2+i)(2-i)} + \frac{5(2+i)}{(2-i)(2+i)} = \frac{5 - 2i}{5} + \frac{10 + 5i}{5} \]

Step 4: Express in Standard Form

Express the result in standard form (the a + bi form). \[1 - \frac{2}{5}i + 2 + i = 3 + \frac{3}{5}i\]