This week in Pre Calc 11, I learned how to solve word problems using quadratic equations and the vertex.
Let’s try an example,
Two numbers have a difference of 6 and their product is a minimum. Determine the numbers.
So first of all, we need to create an equation and set the two numbers values. So, we can set the first number to equal x, and we can have the second number equal x-6. To make an equation out of this it can be x(x-6). The first thing we are going to do is distribute which we get 
. From here, we can pretend the third number is equal to 0, from this we get 
. Next, we need to complete the square. The first step of completing the square is dividing the second term by 2, and squaring it. This comes out to 
. Next we get 
. From here we have to factor the 3 terms, which gets 
. From this equation we get the vertex, which is (3,-9). We get our first number of the two numbers, which is 3. We can put the 3 back into our original equation and get the second number of the two. So, this would be 3-6, which is just equal to -3. Which is our second number.
Let’s try another harder example,
A rectangular play area is to be bounded by 120 m of fencing. Determine the maximum area and the dimensions of this rectangle.
In this question, we are trying to find the area, which is just length times width. We need to find an equation to do this. I have 
and this is all divided by 2. So, we know by this that the length is 60 – w. We now have an equation to solve this. Again, we make the first number w, and the second one 60 – w. So our equation is w(60 – w), and this comes out to 
. To rearrange this it would look like 
. So next, we want to make the 
positive, so we can divide all of the terms by -1. This comes out to 
. From here, we need to complete the square. Which comes out to 
. From here we need to take the -900 out of the bracket and the – sign will affect this and make it positive. This is what it will look like 
. Next, we need to factor the 3 terms. This comes out to 
. This now gives us our vertex. The vertex is 30, 900. We can put the 30 back into our original equation to get our length. So we have 120 – 2(30) all divided by 2. Which will equal 30.
