Video Transcript
If 𝑟 equals negative two plus four 𝑖 and 𝑠 equals eight minus 𝑖, find 𝑟 minus 𝑠 all squared.
In this question, we are given two complex numbers 𝑟 and 𝑠. To find an expression for 𝑟 minus 𝑠, we simply group or collect the real and imaginary parts. Negative two minus eight is equal to negative 10. Four 𝑖 minus negative 𝑖 is equal to five 𝑖. 𝑟 minus 𝑠 is therefore equal to negative 10 plus five 𝑖. We need to square this expression. This involves multiplying negative 10 plus five 𝑖 by negative 10 plus five 𝑖. We can distribute the parentheses or expand the brackets here using the FOIL method. Multiplying negative 10 and negative 10 gives us 100.
Multiplying the outer terms gives us negative 50𝑖. The product of the inner terms is also negative 50𝑖. Finally, multiplying the last terms gives us 25𝑖 squared. 𝑟 minus 𝑠 all squared is equal to 100 minus 50𝑖 minus 50𝑖 plus 25𝑖 squared. Using our knowledge of imaginary numbers, we know that 𝑖 squared is equal to negative one. This means that our expression simplifies to 100 minus 100𝑖 plus 25 multiplied by negative one. This is equal to 75 minus 100𝑖. If 𝑟 is equal to negative two plus four 𝑖 and 𝑠 is equal to eight minus 𝑖, then 𝑟 minus 𝑠 all squared is equal to 75 minus 100𝑖.