Problem 24 - Algebra and Trigonometry

Question

Divide and express the result in standard form. $$ \frac{5 i}{2-i} $$

Step-by-Step Solution

Answer

The division results in the standard form expression, \( -1 + 2i \).

Step 1: Write down the expression

Original expression is \( \frac{5i}{2-i} \)

Step 2: Multiply with conjugate of the denominator

To clear out the imaginary part from the denominator, multiply the expression by the conjugate of the denominator. This means multiplying the numerator and denominator by \(2 + i\). The expression becomes \( \frac{5i(2 + i)}{(2 - i)(2 + i)} \)

Step 3: Simplify the numerator

Expanding the multiplication in the numerator: \(5i*2 + 5i*i = 10i +5i^2 = 10i - 5\) (as \(i^2 = -1\))

Step 4: Simplifying the Denominator

Expanding the multiplication in the denominator: \( (2 - i)(2 + i) = 4 + 2i -2i - i^2 = 4 +1 = 5 \) (as \(i^2 = -1\))

Step 5: Calculate final expression

To obtain the final result, we divide each term in the simplified numerator by the denominator. So, our expression becomes \( \frac{-5}{5} + \frac{10i}{5} = -1 + 2i \)