Question

How do you simplify \[\left( 4+2i \right)\left( -4+2i \right)\]?

Answer 1

Hint: we can simplify these factors using Foil method. First we have to multiply the factors using the foil method. After that we need to perform certain calculations on imaginary terms to get to the result of the equation.

Complete step by step answer:
Let us know about the Foil method.
Foil method (First outer inner last) that is first we have to multiply the first two terms of the factors after that we have to multiply first term of first one and second term of second one after that we have to multiply the second term of first one with first one and then with second one.
The general form of the foil method looks like
\[\left( a+b \right)\left( c+d \right)=ac+ad+bc+bd\]

Now let's solve the given problem
\[\left( 4+2i \right)\left( -4+2i \right)\]
As we can see that the question is in the form of \[\left( a+b \right)\left( c+d \right)\] so we can apply the Foil method.
By applying foil method we will get the equation like this
\[\Rightarrow -16+8i-8i+4{{i}^{2}}\]
As we can see that there are \[+8i\] and \[-8i\]. So we can make them zero by subtracting.
Then we will get
\[\Rightarrow -16+4{{i}^{2}}\]
Here we have \[{{i}^{2}}\] in the above equation.
As we already know that \[i\] will be used to represent \[\sqrt{-1}\].we have \[{{i}^{2}}\] from that we can write it as
\[{{i}^{2}}={{\left( \sqrt{-1} \right)}^{2}}=-1\]
Now we have substitute \[-1\] in place of \[{{i}^{2}}\].
Then we will get
\[\Rightarrow -16+4\left( -1 \right)\]
By simplifying it we will get
\[\Rightarrow -16-4\]
Here we can see both the terms have negative signs so we can add both the terms.
We will get
\[\Rightarrow -20\]
So by simplifying \[\left( 4+2i \right)\left( -4+2i \right)\] we will get \[-20\].
\[\left( 4+2i \right)\left( -4+2i \right)=-20\]

Note: we can also simplify the equation by taking \[2\] as common in two terms and then solve the problem. It will help to reduce the calculation part. Also if we are not aware of the Foil method we can directly multiply both the terms as we multiply numbers.