Question

express in simplest ( a + bi ) form: ((1 - 2xi)(1 + 4i) - (1 - 2xi)(1 + 6i))

Explanation:

Step1: Factor out the common term

Notice that both terms have a common factor of \((1 - 2xi)\), so we can factor it out:

$$ (1 - 2xi)[(1 + 4i) - (1 + 6i)] $$

Step2: Simplify the expression inside the brackets

Simplify \((1 + 4i) - (1 + 6i)\):

$$ (1 + 4i) - (1 + 6i)=1 + 4i - 1 - 6i=(1 - 1)+(4i - 6i)= - 2i $$

Step3: Multiply the factored term with the simplified bracket

Now we have \((1 - 2xi)(-2i)\). Distribute \(-2i\) across the terms in the parentheses:

$$ (1 - 2xi)(-2i)=1\times(-2i)-2xi\times(-2i)=-2i + 4xi^{2} $$

Step4: Substitute \(i^{2}=-1\)

Since \(i^{2}=-1\), substitute this into the expression:

$$ -2i + 4xi^{2}=-2i + 4x(-1)=-4x-2i $$

Answer:

\(-4x - 2i\)