Problem 21 - Algebra and Trigonometry

Question

Divide and express the result in standard form. $$ \frac{2}{3-i} $$

Step-by-Step Solution

Answer

The result in standard form is \(\frac{3}{4} + \frac{i}{4}\).

Step 1: Determine the Complex Conjugate

Given the complex number \(3 - i\), the complex conjugate is \(3 + i\). The complex conjugate of a number \(a + bi\) is \(a - bi\). This switches the sign of the imaginary part of the number.

Step 2: Multiply Numerator and Denominator

Multiply both the numerator and denominator by the complex conjugate from Step 1. \[\frac{2(3 + i)}{(3 - i)(3 + i)}\] This simplifies the fraction since the denominator becomes a real number.

Step 3: Simplify the Numerator and Denominator

Simplify the numerator and denominator. The denominator is simplified using the formula \((a-b)(a+b) = a^2- b^2\). The numerator results in \(6 + 2i\), and the denominator is \((3^2 - (-i)^2) = 9 - 1 = 8\). So the fraction becomes \(\frac{6 + 2i}{8}\).

Step 4: Simplify to Get Standard Form

The final step is to simplify the fraction by dividing each term by the denominator: \(\frac{6}{8} + \frac{2}{8}i = \frac{3}{4} + \frac{i}{4}\).