how do we complete the square?

when completing the square you would usually have an quadratic equation. For exaple x^2 + 8x + __ = 0. you would have to find a number to complete the square. Your first step would be to divide 8 by 2. You would get 4. Then you would square 4 and get 16. So 16 would be the number that would complete the square.

How do we use complex conjugates to divide imaginary numbers?

The conjugate of an expression is you basically changing the operation. For example the conjugate of 4x+3i would be 4x-3i. Now when dividing imaginary numbers using complex conjugates you would take the conjugate of the denominator and multiply the top and bottom by it. The rest is simple you would foil both the top and bottom but when you get to your final answer you would express it in a+bi form. So for example if you get an answer that is 1 take away 21i over 13  you would simplify further and your final answer would be 1 over 13 take away 21 over 13i.

How do we use imaginary numbers?

Imaginary numbers are common in math when using negative numbers in some situations. For example taking the square root of an negative number. We define i equal to the square root of negative 1, also called the imaginary unit. We call i the solution to the equation i squared plus 1 equal 0, or i squared equal negative 1. i to the first power equals i. i to the second power equals -1. i to the third power equals -i. And i to the forth power equals 1. For example taking the square root of -25. The negative sign automatically becomes an i so you can take that out. Now you can factor out 25 which is 5 times 5. The 5 comes out of the radical sign so now you have an i and a 5 out. Your answer would now be 5i. The number always comes before the i.