How do we use imaginary numbers?

               When using the quadratic formula to solve quadratic equations we sometimes run into problems. When finding the discriminant (√b²-4ac) that number can come out to be negative. Here's an example:

      How are you supposed to find the square root of -4? Impossible, right? You may be thinking "Oh. There must be no solution." But there is and using imaginary numbers you can find that solution. An imaginary number is represented by  ,and will help you find your answer.  is the answer to the √-1. This allows us to find the square root of a negative number. Hurray! Lets apply this to √-4.

     First we get rid of the √-1 from √-4 this will leave you with √4. Getting the gist? The problem becomes a breeze from here. Because you know 4 is a perfect square you just simplify and get 2. Don't forget about your imaginary component. The  goes after all real numbers, and before square roots. Making your final answer 2. Piece of cake huh? That my children, is how you use imaginary numbers when solving a quadratic equation. 

What's the point of flipping an inequality symbol?

Why does the inequality sign change when both sides are multiplied or divided by a negative number you ask? The answer is quite simple. The purpose of an inequality is to state that one number is less than (or greater than) another. Now, picture a number line like this:              Now lets use 2 as an example.  When you multiply or divide any number other than zero by a negative, its sign will change. If it was positive it becomes negative and if it was negative it becomes positive. So positive 2 divided by lets say -1, will now give you -2. Well now instead of being on the right sign of the number line the number "flips" and is on the left.                When you multiply or divide both sides of an inequality this happens as well.Both sides "flip" making a number that used to be positive, now negative or vise-versa. So a number can go from being REALLY BIG to teeny tiny. What was less than before is now greater than. This, my children, is why we have to flip the sign whenever we multiply or divide by a negative number. Flipping the sign keeps the inequality making sense.