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Quadratic function

A function of second degree is written as

 


Explanation

You can factorize a quadratic function. Often you can find the solution with little effort, because usually applies a = 1. That you can see clearly

If the solution is not easy you can use a formula. That we are going to develop here, and start with the general form

We separate a with brackets

For the square we are going to prepare a binomial formula

and write this as

The term with an x is handled seperately

Here you can find two solutions

 

and

 

and

 

and

 


Alternative

If the value of b is large, you can develop an alternative formulas, and then you get


and


If you see this it all looks pretty simple. Now we can also calculate

and

That we knew of course already. With this we can calculate the x coordinate of the top, because that is exactly in the middle between x1 and x2.

 


Example 1

This gives (x − 3)(x + 5) = 0 but we could have found this by trial and error. The alternative formulas give

In addition, we see that

        or, using the formula     

         or, using the formula      

     from that follws

There are many different ways to calculate the coordinates of the top. It depends on how far you are already advanced in mathematics. To complete the picture is here the explanation.

In the example y = x2 + 2x + 15 you find the by calculating the lowest value of the function. You can do that by serating a square

A square is always positive, and the lowest value is 0. That occurs at x = –1 and then y = –16. You see that right away.

You can also apply differentiation. With it you can determine the rate of rise of a continuous function. This is the tangent. If that is exactly horizontal, you have the top found. We take again the function

and determine the first derivative

The tangent has the value 0 in the top, so

This is by far the easiest way.

 


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