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Calculate zeros

How can you Calculate zeros? This is exactly what we'll look at in the next few sections. The following content is offered:

  • First of all there is Explanationswhat zeros are and what options there are to calculate zeros.
  • It will Examples pre-calculated to show the various methods such as ABC formula, PQ formula and polynomial division.
  • tasks and Exercises allow you to practice calculating zeros.
  • A Video Calculate for zeroing is also offered.
  • A Question and answer area answers typical questions about finding zeros.

First I will explain very briefly what zeros are. Then it is about which types of functions or equations there are and which method can be used to calculate the zeros of these. Among other things, you will get to know the PQ formula, the midnight formula and the polynomial division. If you have problems with the content, you may be missing a few important prior knowledge: In this case, please deal with the topics of solving equations and drawing functions.

Explanation: Calculate zeros

Before we look at calculating zeros, the following question should be answered: What are zeros? Well, the mathematical description is: a number x0 is called the zero of f if f (x0) = 0. Sounds complicated, doesn't it? So let's take a clearer path. You can plot functions or equations in a coordinate system. This is shown in blue in the next graphic. If you follow its course, you can see that there is a point where it runs through the x-axis. Right here we have the zero (drawn in red). And - look again at the graphic - this is exactly where y = 0.

Fig. 1: Linear equation (function)

A function or equation can of course have more than one zero. This can be seen in the next graphic, where we see a quadratic equation / function that has two zeros (circled in red).

Figure 2: Quadratic equation / function

  • A function or equation can have one or more zeros. Or none at all.
  • We find this out by setting the equation or function to zero, so using f (x) = y = 0.


Zeroing: examples and formulas

How can one calculate zeros? To do this, let's look at numerous examples and corresponding formulas here. The plan looks like this:

How to calculate zeros:

  1. Find out what type of equation or function we have.
  2. Find the appropriate formula or solution method.
  3. Use this formula or method to calculate the zeros.

It doesn't help: we now have to look at what types of equations or functions there are. So that we can then decide which solution method we can use.

Zero point for linear function:

Start with linear equations or linear functions. These have the form:

Examples of linear equations:

  • The highest power in a linear equation is 1.
  • So we only have x here and no x2, x3, x4 or even higher.

Linear equation example 1:

Where is the zero of the equation y = x - 2? Solution: We know that we have to set y = 0 to find the zero.

So we have a zero at x = 2. And this point is characterized by the fact that y = 0 here. So the point of the zero is P (2; 0).

Linear equation example 2:

Where is the zero in the equation y = 4x - 4? Solution: Here, too, we set y = 0 and then calculate x.

The zero point is at x = 1. We know that y = 0 here too. Therefore the point of the zero is P (1; 0).

Zeros quadratic equation / function:

We come to calculating zeros for quadratic functions or quadratic equations. Quadratic equations are of the form:

Examples of quadratic equations:


We now know what quadratic equations are. Just how do you solve this? There are two common methods for doing this. On the one hand there is the PQ formula. On the other hand, there is the ABC formula, which is sometimes also called the midnight formula. With the PQ formula or ABC formula, quadratic functions can be solved (relatively easily). So that you can see how this works, I calculate the task 3x2 + 9x + 5 = - 1 with both variants.

Quadratic equation example 1 (with PQ formula):

Before we can use the PQ formula, you should of course first know what the PQ formula actually looks like. In order to be able to use this one must first ensure that before x2 a 1 stands and the equation is brought to the form with = 0. Then you can read off p and q and simply insert them. First the solution equation, then the example.

We wanted the example 3x2 + 9x + 5 = - solve 1 to calculate the zeros:

  • We know that we need the equation in the form = 0, to do this we first remove the -1 on the right-hand side.
  • We also need before the x2 a 1, i.e. 1x2 and not 3x like here2. So we divide by 3.
  • Then we can simply read p and q and insert them into the solution formula from the last graphic.
  • We calculate the numbers in front of the root and below the root.
  • There is a plus (+) and a minus (-) in front of the root. We calculate x1 with the plus and x2 with the minus.
  • This gives us two solutions. These are the two zeros.

Do you need more examples and explanations for the PQ formula? Then have a look at our article PQ formula.

Quadratic equation example 2 (with ABC formula):

The equation - which we just solved with the PQ formula - should now be solved with the midnight formula. First we transform the equation so that we have = 0. We read off a, b and c and insert them into the solution equation of the ABC formula (midnight formula).

As you can see: PQ formula and ABC formula give the same results.

Cubic functions / 3rd degree, 4th degree or higher function:

Linear functions had an x ​​with them, with quadratic functions the highest power was at x2 reached. And what do I do now when I x3, x4 or even higher? Then we need the polynomial division. Because with the polynomial division we can solve functions of 3rd degree, functions of 4th degree or even higher.

Polynomial division is made up of two words: polynomial and division. You already know divisions from elementary school, for example 6 divided by 2 is a division. Or a fraction with numerator and denominator represents a division. Are we still missing a polynomial: A polynomial is a sum of multiples of powers with natural-number exponents of a variable, which in most cases is denoted by x.

Examples of polynomials:

  • 2x2 + 5x + 8
  • 9x3 + x2 + 5x -3
  • 18x5 + 30x4 + 3x

With the polynomial division, we divide two polynomials by one another. The procedure for calculating zeros looks like this:

  • We need a function or equation whose zeros we want to calculate.
  • We already need a first zero of this function
  • The polynomial division can be carried out with this first zero.
  • You have to guess or try to find out the first zero. Yes, you read that right: Guess and try. At school, however, the teacher usually gives you the first zero.
  • The polynomial division is very similar to the written division. Since many pupils no longer learn the written division in elementary school, we will calculate the example of the polynomial division here very slowly and piece by piece.

Example 1 Polynomial Division:

Let's look at an example of polynomial division. Let x be given3 - 6x2 - x + 6 = 0. Where are the zeros? Solution: By guessing, we get a first zero at x = 1. So we divide x3 - 6x2 - x + 6 through x - 1. If I were to insert x = 1 at x - 1, I would get 0 (zero position). So we have to solve the following problem:

First we write this task down:

Now we have to start calculating. This works in such a way that we first have to perform a division. We first calculate x here3 : x. An x is shortened, i.e. x3 : x = x2.

Next we need to multiply. We calculate x2· (X - 1) = x3 - x2. We write the result under x3 - 6x2.

Now let's subtract as follows and get -5x2.

We now pull the -x down from above:

Now the game starts all over again. In other words, we now have to do a division again: -5x2 : x = -5x

Now we multiply again in the other direction: (-5x) · (x-1) = -5x2 + 5x

And again we subtract (see red box in the next picture):

We pull down + 6:

And now we divide again: (-6x): x = -6

And one last time we multiply: (-6) · (x-1) = -6x + 6

Now if we subtract, we see that the result is 0. And from above (counter) there is nothing more to pull down.

We're done with that: The polynomial division gives (x3 -6x2 - x + 6): (x-1) = x2 -5x -6. But now we want to have the zeros (or have you already forgotten that after such a long calculation?). We still have x2 -5x -6 left. We set this to zero (= 0). And then we can apply the PQ formula to it. If you don't know this yet: The PQ formula is explained above.

If we use the PQ formula then we get at x1 = 6 and at x2 = -1 zeros. Before we did a polynomial division. With this we said at the very beginning that there is still a zero at x = 1. So we have at x3 = 1 a third zero.

Zeroing tasks / exercises


Calculate zeros video

PQ formula video

In the next video you can see how the PQ formula works. First of all, it is briefly explained what a quadratic equation / function is and which solution formula is then used. Corresponding examples are calculated.

Next video »

Calculating zeros: questions and answers

In this section we look at typical questions about computing zeros. With appropriate answers.

Q: Should I use the PQ formula or the ABC formula for quadratic functions?

A: Both of them work. I myself find the PQ formula easier, but that's a matter of taste. Is in front of the x2 a 1, then the PQ formula is usually the easier variant. If you need further information on both types, you can also look into the article PQ formula or ABC formula (article will be written shortly and then also linked here).

Q: How do I find the zeros for sine and cosine functions?

A: Finding zeros in functions with sine or cosine is a separate topic. We deal with this in the article Zeroing Sine / Cosine.

Q: How can I practice this topic well?

A: In our article Calculating Zeros: Tasks / Exercises.