Quadratics and Negative Roots

  I've been working on helping my students improve their math literacy and today, I encountered another situation where we choose the answer depending on the situational context.

We were discussing the square root of a number in terms of solving an equation.  I explained in this problem, we need both roots for the solution.

One of the students asked about x^2 in terms of a room.  He wanted to know why we only used the positive root.  This lead to a discussion on rooms using only positive measurements.  So then the question came up when do you use the negative square root?

I had to stop and think a moment.  I know its often used in the standard deviation but I had to do a web search to really answer that question.  I posted my dilemma to Google + Mathematics Education group for help with this.  One of the members pointed out that the negative roots are important in quadratics which lead me to realize that negative numbers are important for parabolic shapes such as mirrors and satellite dishes. 

Then it struck me, the negative is not a subtraction but a direction away from the center of the parabola.  It all relates back to the context of the math.  So now when I introduce my unit on  quadratics, I can reaffirm the negative is a direction. 

I did a quick search of quadratics and found they are applied to the following:

1. Calculating area - You can use it to find the measurement of a room or to find out what your room measurement might be if you know you enlarged the room or made it smaller and knew the area.

2. Finding the rise and fall of profit for a business.

3. Finding the speed of something that has wind or current acting on it.  

4.  Certain parts of vectors.

5. Acceleration.

6. Stopping distance.

7. Air movement.

8.  Aiming missles and rockets.

There are lots more examples but this is a good start.