Simplify three minus six 𝑖 all squared multiplied by two minus 𝑖.
We can begin by rewriting three minus six 𝑖 all squared as three minus six 𝑖 multiplied by three minus six 𝑖. We can distribute the parentheses or expand the brackets here using the FOIL method. Multiplying the first terms gives us nine. Multiplying the outer terms gives us negative 18𝑖. The inner terms also have a product of negative 18𝑖. Finally, multiplying the last terms gives us 36𝑖 squared. Three minus six 𝑖 multiplied by three minus six 𝑖 is equal to nine minus 18𝑖 minus 18𝑖 plus 36𝑖 squared.
We recall from our knowledge of imaginary numbers that 𝑖 squared equals negative one. Therefore, 36𝑖 squared is negative 36. Taking this away from nine gives us negative 27. And negative 18𝑖 minus 18𝑖 is negative 36𝑖. Our expression simplifies to negative 27 minus 36𝑖. We now need to multiply this expression by two minus 𝑖. Once again, we will use the FOIL method. Negative 27 multiplied by two is negative 54. Negative 27 multiplied by negative 𝑖 is 27𝑖. Our next term is negative 72𝑖. And finally, we have 36𝑖 squared. Negative 27 minus 36𝑖 multiplied by two minus 𝑖 is equal to negative 54 plus 27𝑖 minus 72𝑖 plus 36𝑖 squared.
We will once again use the fact that 𝑖 squared is equal to negative one. Grouping the real and imaginary parts gives us negative 100 minus 45𝑖. Three minus six 𝑖 all squared multiplied by two minus 𝑖 is equal to negative 100 minus 45𝑖.