Problem 20 - Algebra and Trigonometry

Question

Find each product and write the result in standard form. $$ (5-2 i)^{2} $$

Step-by-Step Solution

Answer

The square of the complex number \(5 - 2i\) is \(21 - 20i\)

Step 1: Expanding the Expressions

Firstly, square the complex number by expanding \( (5-2i)^2\) as \( (5-2i)(5-2i) \). Then we will use the distributive property and multiply each term in the first brackets with every term in the second brackets.

Step 2: Perform Multiplications

Perform the multiplication as follows: Start by multiplying real part with real part: \(5 * 5 = 25\), then continue multiplying real part with imaginary part and imaginary part with real part: \(5 * -2i + -2i * 5 = -10i -10i = -20i\), and finally we have to multiply the imaginary parts: \(-2i * -2i = 4i^2\). Here, remember that \(i^2\) is defined as -1 in complex numbers.

Step 3: Combine like terms

Now, combine all of the results from the previous step which are \(25, -20i\) and \(4i^2\). This gives us \(25 -20i + 4(-1)\) which simplifies to \(25 -20i - 4\).

Step 4: Simplify Final Answer

Simplify the expression by combining real numbers (25 and -4) to yield the final answer \(21 - 20i\) in standard form for complex numbers.