How do you find two consecutive even integers whose products is 728?
Represent two consecutive even integers with a little algebra, then solve the subsequent quadratic to get the two numbers of
The first thing we need to do is figure out how to generate an even number.
This is actually a pretty simple task when we think about what an even number is. Consider the first few even numbers:
We can see right off the bat that they're all multiples of
For example, if we want to find the 20th even number, we could write down the first 20...
...or we could use our neat little formula,
In both cases, we see that the 20th even number is 40.
Now, how about consecutive even numbers? Well, let's start out with some general even number and call it
The problem asks us to find the product of two consecutive even integers (and an integer is a fancy word for whole negative/positive number, like
Multiplying out, we have:
Subtracting 728 from both sides gives us:
I wouldn't want to factor this equation unless I had to (i.e. my teacher forced me), so using some technology, we find that the roots of this equation are:
Even though both of these are valid solutions, we'll go with the positive solution of
So, what does it mean? It means that the 13th even number times the even number after it (the 14th even number) equals 728. Let's first find out what these two numbers actually are, using our neat formulas:
If you multiply