Question

simplify the expression to a + bi form: (4 - i)(2 - 3i)

Explanation:

Step1: Use the distributive property (FOIL method)

Multiply each term in the first complex number by each term in the second complex number:
$$(4 - i)(2 - 3i)=4\times2+4\times(-3i)-i\times2+(-i)\times(-3i)$$

Step2: Simplify each product

Calculate each product:
$$4\times2 = 8$$
$$4\times(-3i)=-12i$$
$$-i\times2=-2i$$
$$(-i)\times(-3i) = 3i^{2}$$
So the expression becomes:
$$8-12i - 2i+3i^{2}$$

Step3: Recall that \(i^{2}=-1\)

Substitute \(i^{2}=-1\) into the expression:
$$8-12i - 2i+3\times(-1)$$

Step4: Simplify the real and imaginary parts

Simplify the real numbers and combine like - imaginary terms:
First, simplify the real part: \(8 + 3\times(-1)=8 - 3=5\)
Then, combine the imaginary terms: \(-12i-2i=-14i\)
So the simplified form is \(5-14i\)

Answer:

\(5 - 14i\)