How to Find Median in 3 Easy Steps

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How to Find Median in 3 Easy Steps

How to Find Median in Math Step-by-Step

Math Skills: How to find the median of a data set in 3 easy steps

 

Are you ready to learn how to find the median of a data set?

 

Finding the median of a data set is an important skill that will help you to analyze and make conclusions about data.

In math, the word median means middle and it refers to the middle number in a data set where the values are organized from smallest to largest.

In this Free Step-by-Step Guide on How to Find Median, you will learn how to find the median of a data set using a simple 3-step method, which we will practice and apply to several example problems. This guide will cover the following sections/topics:

You can click on any of the quick-links above to jump to any topic/section of this guide, or you can follow each section in order (we highly recommend this option if you are new to this topic).

Let’s begin!

 

Lesson Preview: How to find median in math in 3 easy steps.

 

What is the Median of a Data Set?

In statistics, a data set is a group of numbers that represent some form of data.

For example, consider a data set that represents the ages of 5 different students:

  • 12, 13, 15, 17, and 20

Note that a values in a data set can be represented as a list of numbers (as shown above) or contained with a set of curly brackets as shown below:

  • {12, 13, 15, 17, 20}

Note that both of these options represent the same data set, which, in this case, represents the ages of students.

Figure 01 below illustrates this data set and what the values represent.

 

Figure 01: This data set represents the ages of five different students.

 

In math, the median of a data set is the middle value of a set of data that is arranged in order from smallest to largest from left to right.

As long as the values in a data set are arranged in order of smallest to largest, you can simply locate the middle value to determine the median of the data set.

You can find the median, or middle, value of any data set by following these three simple steps:

  • Step One: List all of the numbers in order from smallest to largest

  • Step Two: Identify the middle number (if there are two middle numbers, continue to Step Three).

  • Step Three: If there are two middle numbers, find the average of the two numbers to determine the median.

Before we move onto any example problems of how to find the median of a data set, let’s apply these three steps to finding the median of the previously mentioned data set that represented the ages of five students as 12, 13, 15, 17, and 20.

Step One: List all of the numbers in order from smallest to largest

If we wanted to find the median of the data set that represented student’s ages, we would first have to make sure that the values are indeed in order from smallest to largest:

  • 12, 13, 15, 17, 20

It is easy to see that the numbers in the data set are arranged in order from smallest to largest.

Step Two: Identify the middle number (if there are two middle numbers, continue to Step Three).

  • 12, 13, 15, 17, 20

Step Three: If there are two middle numbers, find the average of the two numbers to determine the median.

In this case, the middle number is 15, so we can skip the third step and conclude that:

Final Answer: The median is 15.

Note that, later on in this guide, we will work through example problems where Step Three will be necessary.

The 3-step process for how to find median is displayed in Figure 02 below.

 

Figure 02: The median of a data set with values arranged in order from smallest to largest is the middle value.

 

Now that you are familiar with the concept of a median of a data set and what it represents, you are ready to gain some more practice with how to find median using our 3-step method.

How to Find Median Examples

How to Find the Median of a Data Set Example #1

Example: Find the Median of the Data Set:

Data Set: 3, 9, 11, 5, 17, 8, 16

Step One: List all of the numbers in order from smallest to largest

Notice that the numbers in our data set are not in order from smallest to largest. We can rewrite the data set so that the numbers are indeed in order as follows:

  • 3, 5, 8, 9, 11, 16, 17

Now that the numbers in our data set are arranged in order from smallest to largest, we are ready for our second step.

Step Two: Identify the middle number (if there are two middle numbers, continue to Step Three).

Since our data set has an odd number of values, we pinpoint one exact middle term, which, in this case, is 9:

  • 3, 5, 8, 9, 11, 16, 17

Step Three: If there are two middle numbers, find the average of the two numbers to determine the median.

In cases such as this, we can skip our third step and conclude that:

Final Answer: The median is 9.

Figure 03 below illustrates how we solved this first example step-by-step.

 

Figure 03: How to Find Mean Using 3 Easy Steps.

 

Notice that our data set for this first example has an odd amount of values (the data set had 7 total values). Whenever this is the case, you will be able to identify one single number and skip Step Three.

However, whenever there is an even amount of values, there will be two middle numbers and you will have to perform the third step to determine the median, which will be the case in the next example.

How to Find the Median of a Data Set Example #2

Example: Find the Median of the Data Set:

Data Set: 5, 14, 4, 9, 24, 16, 10, 18

Step One: List all of the numbers in order from smallest to largest

Our first step is to arrange all of the numbers in order from smallest to largest as follows:

  • 4, 5, 9, 10, 14, 16, 18, 24

Step Two: Identify the middle number (if there are two middle numbers, continue to Step Three).

Unlike the last example which had a data set with an odd amount of numbers, the data set in this example has an even amount of numbers (there are 8 total values).

In cases like this, when there is an even amount of numbers in the data set, there will be two middle numbers:

  • 4, 5, 9, 10, 14, 16, 18, 24

Notice that both 10 and 14 are both in the middle. Whenever there are two middle numbers, you will have to complete Step Three to determine the median of the data set.

Step Three: If there are two middle numbers, find the average of the two numbers to determine the median.

To find the median of a data set with two middle numbers, you simply have to find the average of the two numbers. You can do this by adding the two middle numbers together and then dividing by two as follows:

  • 10 + 14 = 24

  • 24 ÷ 2 = 12

Final Answer: The median is 12.

The step-by-step process for how to find a median of a data set with an even amount of values is shown in Figure 04 below.

 

Figure 04: How to Find Median of a Data Set with Two Middle Numbers.

 

Now that you are familiar with how to find median when a given data set has an odd amount of numbers or an even amount of numbers, let’s work through two more examples to further reinforce your understanding of how to find a median.

How to Find the Median of a Data Set Example #3

Example: Find the Median of the Data Set:

Data Set: 2, 58, 40, 17, 15, 16, 31, 6, 10, 18, 67, 23, 96

Step One: List all of the numbers in order from smallest to largest

First, rewrite the numbers as a list in order from smallest to largest:

  • 2, 6, 10, 15, 16, 17, 18, 23, 31, 40, 58, 67, 96

Step Two: Identify the middle number (if there are two middle numbers, continue to Step Three).

Notice that this particular data set has an odd amount of numbers (there are 13 total values). So, we can determine the middle value on the second step:

  • 2, 6, 10, 15, 16, 17, 18, 23, 31, 40, 58, 67, 96

Step Three: If there are two middle numbers, find the average of the two numbers to determine the median.

Since this data set had an odd amount of values, we were able to identify the median in Step Two and we can conclude that:

Final Answer: The median is 18.

Figure 05 below shows how we identified the median in Example #3.

 

Figure 05: How to Find a Median of a Data Set.

 

How to Find the Median of a Data Set Example #4

Example: Find the Median of the Data Set:

Data Set: 2, 9, 5, 8, 4, 10, 3, 15, 1, 25

Step One: List all of the numbers in order from smallest to largest

Just like the previous examples, we begin by rearranging the numbers in the data set so that they are in order from largest to smallest:

  • 1, 2, 3, 4, 5, 8, 9, 10, 15, 25

Step Two: Identify the middle number (if there are two middle numbers, continue to Step Three).

The data set in this Example #4 has an even amount of numbers (there are 10 total values).

Whenever there is an even amount of numbers in the data set, there will be two middle numbers:

  • 1, 2, 3, 4, 5, 8, 9, 10, 15, 25

Both 5 and 8 are in the middle, so we have to move onto Step Three to determine the value of the median.

Step Three: If there are two middle numbers, find the average of the two numbers to determine the median.

We can identify the median by finding the average of the two middle numbers as follows:

  • 5 + 8 = 13

  • 13 ÷ 2 = 6.5

Final Answer: The median is 6.5.

Notice that our final answer is a decimal, which is totally fine and quite common!

 

Figure 06: How to Find a Median Step-by-Step

 

Conclusion: How to Find Mean

Knowing how to find the median of a data set is an important and useful math skill that every student must learn.

The median of a data set that is arranged in order from smallest to largest is the middle value (i.e. median = middle).

You can find a median by applying the following three-step method:

  • Step One: List all of the numbers in order from smallest to largest

  • Step Two: Identify the middle number (if there are two middle numbers, continue to Step Three).

  • Step Three: If there are two middle numbers, find the average of the two numbers to determine the median.

Note that, for data sets with an odd amount of values, Step Three can be skipped. However, whenever a data set has an even amount of values, there will be two middle numbers and following Step Three will be necessary for finding the median value.

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How to Find Mode in Math—Explained

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How to Find Mode in Math—Explained

How to Find the Mode in Math Step-by-Step

Math Skills: How to find mode of a data set

 

Are you ready to learn how to find mode?

 

In math, it is important for students to understand different ways of measuring and describing central tendency when it comes to sets of data. One of the simplest ways to do this is a method called finding the mode, or most common number, of a given data set.

This free step-by-step guide on How to Find Mode will teach you everything you need to know about the mode of a set of numbers, how to find it, and what it represents.

You can use the links below to jump to any section of this guide, or you can work through each section in order.

Ready to get started?

 

Lesson Preview: How to find mode of a given data set.

 

What is the Mode of a Data Set?

In math, the term data set refers to a collection of values that represent some type of data.

For example, the amount of money in savings for five different students is $18, $21, $9, $13, and $21.

These dollar amounts can be expressed as a data set simply by writing the collection of numbers inside of curly brackets (separated by commas) as follows:

  • {18, 21, 9, 13, 21}

This scenario where we are using a data set to represent the savings of five students in illustrated in Figure 01 below.

 

Figure 01: How to Find Mode of a Data Set.

 

In math, the mode of a data set is the value that has the highest frequency. In other words, the mode is the value in the data set that is present more than any other number (i.e. the mode is the most common number).

Note that: a data set can have one mode, no mode, or multiple modes.

While finding the mode of a data set is a simple process, it is an important skill because it will help you to identify trends within a set of data and how it is distributed.

How to Find Mode

To find the mode of a data set, simply look at all of the values in the set and determine which value appears the most. Remember that a data set can have one mode, multiple modes, or no mode (we will see instances of all three scenarios in the examples below).

For example, how can we find the mode of the data set that represented student’s savings?

Remember that the data set included the following values:

  • {18, 21, 9, 13, 21}

In this case, notice that one value is present twice (21)), while all of the other values only occur once.

Therefore, we know that 21 is the most common number in the data set and we can conclude that:

Final Answer: The mode is 21.

The step-by-step process for how to find the mode in math are shown in Figure 02 below.

 

Figure 02: How to Find Mode: Carefully look for the most common value.

 

Now that you know what the mode of a data set represents and how to identify the mode, you are ready to work through several problems where you will gain more experience with identifying the mode of a given set of data.

How to Find the Mode Examples

How to Find the Mode Example #1

Example: What is the mode of the data set?

Data Set: 5, 2, 8, 2, 7, 3, 2

Remember that the mode of a data set is the value with the highest frequency (i.e. the value that occurs the most times).

Given the data set above, only one value occurs more than once, and that value is 2 (which occurs 3 times). So, we can conclude that the data set has a mode of 2.

Final Answer: The mode is 2.

Notice that the data set in this example has one, and only one, mode. Figure 03 below illustrates how to find the mode for this first example.

 

Figure 03: How to Find Mode of a Data Set.

 

Now that we have solved this first example, we can see that determining the mode of a given data set is relatively easy to do.

However, there are a few nuances to finding the mode as you will see in the next few examples.

How to Find Mode Example #2

Example: What is the mode of the data set?

Data Set: 14, 14, 17, 9, 11, 17, 24, 22, 17, 12

For this next example, it may be tempting to say that the mode of the data set is 14 since 14 occurs more than once. However, this is actually not the case. While it is true that 14 occurs twice in this data set, there is a number that occurs more than twice: 17 occurs 3 times.

Since 17 occurs more than any other number (it occurs 3 times), we can conclude that:

Final Answer: The mode is 17.

Just like the previous example, this data set has only one mode.

 

Figure 04: How to Find Mode in Math

 

How to Find Mode Example #3

Example: What is the mode of the data set?

Data Set: 5, 11, 9, 8, 9, 9, 11, 11, 7, 5, 11, 10, 9, 3, 11, 12

Notice that the given data set in this example is much larger than the data sets from the first two examples. In this case, we have several numbers that occur multiple times, so it would be helpful to make a quick tally chart for each value so we can accurately determine which value in this data set occurs the most:

  • 3: once

  • 5: 2 times

  • 7: once

  • 8: once

  • 9: 4 times

  • 10: once

  • 11: 5 times

  • 12: once

Now that we have logged the frequency of each value in the data set, we can see that the number with the highest frequency is 11 (it occurs 5 times).

Final Answer: The mode is 11.

Again, the data set in this example has only one mode.

 

Figure 05: How to Find Mode Explained.

 

How to Find Mode Example #4

Example: What is the mode of the data set?

Data Set: 5, 5, 5, 7, 7, 9, 9, 9, 7, 7, 9

Just like the last example, it is difficult to determine the mode of this data set simply by looking at it, so we will again create a tally chart to keep track of the frequency of each value:

  • 5: 3 times

  • 7: 4 times

  • 9: 4 times

Notice that this data set does not have one single value that occurs more than any other value. In fact, there is a “tie” in this case, where both 7 and 9 occur 4 times each in the data set.

In cases such as this, the data set can be considered multimodal, meaning that it has more than one mode (in this case, the data set has 2 modes).

Final Answer: The mode is 7 and 9.

 

Figure 06: Example of a multimodal data set (i.e. a data set that has more than one mode).

 

How to Find Mode Example #5

Example: What is the mode of the data set?

Data Set: 13, 0, 24, 83, 81, 55, 16, 42, 22

For this final example, we have a data set with nine values. We can track the frequency of each value as follows:

  • 0: once

  • 13: once

  • 16: once

  • 22: once

  • 24: once

  • 42: once

  • 55: once

  • 81: once

  • 83: once

Notice that each value in the data set only occurs once. In cases like this, we have a data set with no mode.

Final Answer: There is no mode.

 

Figure 07: Example of a data set that has no mode.

 

Conclusion: How to Find Mode

Understanding how to find the mode in math is a simple, yet important skill that will help you analyze and draw conclusions about sets of data.

In math, the mode of a data set is the value with the highest frequency (i.e. the value that occurs the most in the data set).

There are three ways to classify the mode of a data set:

  • One Mode: There is one value that occurs more than any other value in the data set.

  • Multimodal: There is more than one value that occurs the most (i.e. there is a “tie” for the highest frequency).

  • No Mode: Each value in the data set occurs only once.

With these nuances in mind, determining the mode of a data set is a pretty simple skill that all students can master with some study and practice.

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How to Find Mean in 3 Easy Steps

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How to Find Mean in 3 Easy Steps

How to Find the Mean in Math Step-by-Step

Math Skills: How to find the mean of a data set in 3 easy steps

 
 

Understanding how to find mean is an important and fundamental math skill that every student must learn at one point or another.

In math, the term mean is synonymous with average, meaning that the mean of a data set represents the average value of that data set.

This free guide will teach you the mathematical definition of mean and how to find the mean of a data set using a simple 3-step process. The following topics related to how to find mean will be covered:

While you can use the hyperlinks above to jump to any section of this guide, we highly recommend that you follow each section in order, starting with the first section below, where we will review the definition of mean and what it means in relation to a given data set.

Let’s get started!

 

Lesson Preview: How to find the mean in math in 3 easy steps.

 

What is the Mean of a Data Set?

In math, a data set is a collection or group of data in the form of numbers.

For example, consider the following data set which represents the saving amounts, in dollars, of five high school students:

  • $20, $25, $5, $25, and $15

These savings amounts can be expressed as a data set by removing the “$” signs and writing the numbers inside of curly brackets as follows:

  • {20, 25, 5, 25, 15}

This example of expressing the savings amounts of five high school students is illustrated in Figure 01 below.

 

Figure 01: What is a data set in math?

 

In math, the mean of a data set is one single value that represents the average of the entire data set.

In other words, the mean is a measure of central tendency (i.e. the average of a given data set). The mean can be found by taking the sum of all of the numbers in a data set and dividing that value by the total amount of data points.

So, you can find the mean of any data set by adding up all of the numbers and dividing that sum by the total amount of numbers in the data set.

How to Find Mean in 3 Easy Steps:

  • Step One: Add together all the numbers in the data set to find the sum.

  • Step Two: Determine how many numbers are in the data set.

  • Step Three: Divide the sum by the total amount of numbers in the data set to determine the mean.

To find the mean of the data set that represented student savings (as shown in Figure 01 above), we can simply follow these three steps to find the mean as follows:

Find the Mean of: {20, 25, 5, 25, 15}

Step One: Add together all the numbers in the data set to find the sum.

We can start by finding the sum of all of the numbers in the data set as follows:

  • 20 + 25 + 5 + 25 + 15 = 90

Step Two: Determine how many numbers are in the data set.

  • This particular data set has 5 numbers.

Step Three: Divide the sum by the total amount of numbers in the data set to determine the mean.

Finally, we just have to take our result from step one (the sum of all of the numbers in the data set) and divide it by the result of step two (the total number of numbers in the data set) as follows:

  • 90 ÷ 5 = 18

So, we can conclude that the average savings amount per student is $18.

Final Answer: The mean is 18.

The step-by-step process for how to find the mean in math are shown in Figure 02 below.

 

Figure 02: You can find the mean of any data set by adding up all of the numbers and dividing the sum by the total amount of numbers in the set.

 

Now that you understand what the mean of a data set represents and how to find it, let’s use our three-step process to solve three examples of how to find the mean of a data set.

How to Find the Mean of a Data Set

How to Find Mean Example #1

Example: Find the Mean of a Group of Siblings’ Ages

In this first example, we have a data set that represents the ages of six siblings with the following ages: 20, 22, 24, 26, 28, and 30.

So, our data set can be expressed as: {20, 22, 24, 26, 28, 30)

Step One: Add together all the numbers in the data set to find the sum.

Start by adding all of the numbers in the data set together as follows:

  • 20 + 22 + 24 + 26 + 28 + 30 = 150

Step Two: Determine how many numbers are in the data set.

  • The data set in this example has 6 numbers.

Step Three: Divide the sum by the total amount of numbers in the data set to determine the mean.

Now we are ready to find the mean by dividing the sum of all of the numbers by the total amount of numbers as follows:

  • 150 ÷ 6 = 25

Now we can conclude that the mean age of the siblings is 25 years old.

Final Answer: The mean is 25.

The entire step-by-step process for how to find the mean for this first example is illustrated in Figure 03 below.

 

Figure 03: How to Find Mean Using 3 Easy Steps.

 

As you can see from this first example, the process of finding the mean of a data set is an easy process that can be completed using just three steps.

The more that you practice using these steps on how to find mean, the easier these types of problems will become. Now, let’s move onto another example of how to find the mean of a data set.

How to Find the Mean in Math

How to Find Mean Example #2

Example: Find the Mean of a Data Set Representing Monthly Income

In this example, John has been tracking his monthly income and recorded the data in the following data set:

  • $1,200, $1,350, $1,250, $1,400, $1,300, $1,000, $900

For this example, we can make the values easier to work with by removing the dollar signs and expressing the values in a data set as follows:

  • {1200, 1350, 1250, 1400, 1300, 1000, 900}

Now we can use our 3-step process for how to find mean as follows:

Step One: Add together all the numbers in the data set to find the sum.

Just like the previous example, we can start by finding the sum of all of the numbers in the data set:

  • 1200 + 1350 + 1250 + 1400 + 1300 + 1000 + 900 = 8400

Step Two: Determine how many numbers are in the data set.

  • This data set has 7 numbers.

Step Three: Divide the sum by the total amount of numbers in the data set to determine the mean.

The last step is to divide the sum of all of the values in the data set by the total number of values in the data set:

  • 8400 ÷ 7 = 1,200

Finally, we can conclude that the mean monthly income is $1,200.

Final Answer: The mean is 1,200.

The process for finding the solution to this second example is shown in Figure 04 below.

 

Figure 04: How to Find the Mean of a Data Set.

 

Notice how our final answers in our first two examples ended up being whole numbers. This, however, will not always be the case. Now, let’s work through one more example where our final result will be a decimal.

How to Find the Mean of a Data Set

How to Find Mean Example #3

Example: Find the Mean of a Data Set Representing Student Test Scores

In this third example, we are looking to find the mean (or average) test score of a student with the following test scores: 78, 85, 92, 88, 80, 63, 90, 89, 71, and 95

Let’s go ahead and use our 3-step method for how to find mean of a data set to solve this problem and determine the mean test score.

Step One: Add together all the numbers in the data set to find the sum.

Our first step is to find the sum of all of the numbers in the data set:

  • 78 + 85 + 92 + 88 + 80 + 63 + 90 + 89 + 71 + 95 = 831

Step Two: Determine how many numbers are in the data set.

  • There are 10 values (or test scores) in this data set.

Step Three: Divide the sum by the total amount of numbers in the data set to determine the mean.

Finally, we just have to take our result from step one (831) and divide it by the total number of values in the data set (10) as follows:

  • 831 ÷ 10 = 83.1

Now we can make the conclusion that the student had an average test score of 83.1

Final Answer: The mean is 83.1

Notice how our result for this example was a decimal, which is totally fine!

Figure 05 below illustrates how we solved this third example.

 

Figure 05: How to Find Mean with Decimal Answers.

 

How to Find Mean Challenge

Challenge Problem

Scenario: On their last five science exams, students had the following scores:

  • Bonnie: 83, 100, 95, 61, 68

  • Celeste: 88, 83, 79, 86, 90

  • Michelle: 97, 64, 60, 89, 73

Which student had the highest mean test score?

For this challenge problem, you are tasked with determining which of the three students had the highest mean (or average) test score.

To solve this problem, you will have to calculate the mean test score of each student and then compare the means to see which on was the highest value.

 

Figure 06: Which student had the highest mean test score?

 

We can find the mean test score for each student by adding all five of their scores together and then dividing that sum by 5 as follows:

  • Bonnie: 83 + 100 + 95 + 61 + 68 = 407 → 407 ÷ 5 = 81.4

  • Celeste: 88 + 83 + 79 + 86 + 90 = 426 → 426 ÷ 5 = 85.2

  • Michelle: 97 + 64 + 60 + 89 + 73 = 393 → 393 ÷ 5 = 78.6

Now we can conclude that:

  • Bonnie’s mean test score was 81.4

  • Celeste’s mean test score was 85.2

  • Michelle’s mean test score was 78.6

And, we can conclude that Celeste’s 85.2 mean test score was the highest of the three students.

Final Answer: Celeste had the highest mean test score, which was 85.2.

 

Figure 07: How to Find Mean in Math: Test Scores

 

Conclusion: How to Find Mean

Knowing how to find the mean of a data set is an important math skill that every student should be familiar with.

In math, a data set is a collection of data values and the mean of a data set represents the average value of that set. You can find the mean of any data set by following the following three steps:

  • Step One: Add together all the numbers in the data set to find the sum.

  • Step Two: Determine how many numbers are in the data set.

  • Step Three: Divide the sum by the total amount of numbers in the data set to determine the mean.

Once you gain enough practice with finding the mean, you can simplify this process for how to find the mean by simply adding together all of the numbers in the data set and then dividing that sum by the total number of values.

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How to Complete the Square in 3 Easy Steps

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How to Complete the Square in 3 Easy Steps

How to Complete the Square in 3 Easy Steps

Step-by-Step Guide: How to do Completing the Square

 

Step-by-Step Guide: How to Complete the Square Explained.

 

As you continue onto more advanced problems where you have to factor quadratics, you will have to learn how to complete the square in order to find correct solutions. Completing the square is a special technique that you can use to factor quadratic functions.

This free step-by-step guide on How to Complete the Square will teach you an easy 3-step method for factoring any quadratic using a technique called “completing the square.”

This guide will focus on the following topics and sections. You can click on any of the text links below to jump to one particular section, or you can follow each section in sequential order.

Let’s begin by exploring the meaning of completing the square and when you can use it to help you to factor a quadratic function.

 

Figure 01: Are you ready to learn how to find solutions to quadratic equations by completing the square?

 

What is Completing the Square?

Completing the square is a method that you can use to solve quadratic equations of the form ax² + bx + c = 0 (where a, b, and c are all not equal to zero).

❗Note that the equations of the form ax² + bx + c = 0 are called quadratic equations and they can be rewritten as follows:

  • ax² + bx + c = 0

  • ax² + bx = -c

Both of these equations are equivalent to each other, and understanding the relationship between these two equations will help you to understand how to complete the square later on in this guide.

In the next section, we will work through three examples of how to complete the square using the following 3-step method:

  • Step #1: Rearrange the quadratic equation so that all of the constants are on one side of the equals sign.

  • Step #2: Add (b/2)² to both sides of the equal sign.

  • Step #3: Factor and solve.

By solving a quadratic equation by completing the square, you are identifying values where the parabola that represents the equation crosses the x-axis.

As long as you understand how to follow and apply these three steps, you will be able to solve quadratics by completing the square (provided that they are solvable). Now, let’s gain some experience with using the three step method on how to complete the square by working through some step-by-step practice problems.

 

Figure 02: The solutions to a quadratic equation are the values where the graph crosses the x-axis.

 

How to Complete the Square: Example #1

Solve: x² - 6x -16 = 0

For this first example (and all of the practice problems in this guide), we can solve the problem by completing the square using our three step method as follows:

Step #1: Rearrange the quadratic equation so that all of the constants are on one side of the equals sign.

Let’s start off by noticing that our given quadratic function is indeed in ax² +bx + c = 0 form, where a=1, b=-6, and c=-16.

To complete the first step, we have to move all of the constants (all of the values not attached to variables, to the right side of the equals sign as follows:

  • x² - 6x -16 = 0

  • x² - 6x -16 (+16) = 0 (+16)

  • x² - 6x = 16

Now we have completed the first step and we are left with a new equivalent equation:

  • x² - 6x = 16

Step #2: Add (b/2)² to both sides of the equal sign.

For the next step, we have to find the value of (b/2)² and add it to both sides of the equals sign.

Since we know that b=-6, we can find the value of (b/2)² by substituting -6 for b as follows:

  • (b/2)²

  • (-6/2)²

  • (-3)²

  • =9

In this case, (b/2)² = 9, so, to complete Step #2, we simply have to add 9 to both sides of the equal sign as follows:

  • x² - 6x = 16

  • x² - 6x + 9 = 16 + 9

  • x² - 6x + 9 = 25

 

Figure 03: How to Complete the Square in 3 Easy Steps.

 

Step #3: Factor and solve.

Finally, we are ready for the third and final step where we just need to factor and solve.

Notice that the left side of the equation of x² - 6x + 9 = 25 is a trinomial that is factorable as follows:

  • x² - 6x + 9 = (x-3)(x-3)

  • x² - 6x + 9 = (x-3)²

In this example, the factors of x² - 6x + 9 are (x-3)(x-3), which we will express as (x-3)² since it will allow us to solve the problem as follows:

  • x² - 6x + 9 = 25

  • (x-3)² = 25

  • √[(x-3)²] = √[25]

  • x -3 = ± 5

In the third step above, we took the square root of both sides of the equation to remove the exponent and we are left with x -3 = ± 5, which means that

  • x - 3 = 5

  • x - 3 = -5

We can now find the solutions to this first example by solving both equations as follows:

  • x - 3 = 5 → x = 8

  • x - 3 = -5 → x = -2

All three steps for how to do completing the square are shown in Figure 03 above.

Now, we can conclude that the original quadratic equation x² - 6x -16 = 0 has two solutions:

Final Answer: x = 8 and x = -2

This means that the graph of the equation x² - 6x -16 = 0 will be a parabola that crosses the x-axis at both (-2,0) and (8,0) as shown in Figure 04 below.

 

Figure 04: How to solve by completing the square: graph explanation

 

How to Complete the Square: Example #2

Solve: x² +12x +32 = 0

We can solve this next example using our 3-step method, just as we did in the previous example, as follows:

Step #1: Rearrange the quadratic equation so that all of the constants are on one side of the equals sign.

Notice for our given quadratic equation, x² +12x +32 = 0, that a=1, b=12, and c=32.

Since our constant c is on the left side of the equation, we simply have to move it to the right side using inverse operations to complete Step #1.

  • x² +12x + 32 = 0

  • x² +12x + 32 (-32) = 0 (-32)

  • x² + 12x = -32

After completing the first step, we now have:

  • x² + 12x = -32

Step #2: Add (b/2)² to both sides of the equal sign.

Next, we have to add (b/2)² to both sides of our new equation.

In this example, b=12, so we can find the value of (b/2)² as follows:

  • (b/2)²

  • (12/2)²

  • (6)²

  • =36

Since (b/2)² = 36, we can complete Step #2 by adding 36 to both sides of the equation as follows:

  • x² + 12x = -32

  • x² + 12x +36 = -32 +36

  • x² + 12x +36 = 4

 

Figure 05: How to complete the square to find the solutions to a quadratic equation.

 

Step #3: Factor and solve.

For the final step, we just have to factor and solve for any potential values of x.

Just like example #1, we can finish completing the square by factoring the trinomial on the left side of the equation and then solving.

In this case, the trinomial on the left side of the equation can be factored as follows:

  • x² + 12x +36 = (x+6)(x+6)

  • x² + 12x +36 = (x+6)²

❗Note that whenever you solve a problem using the complete the square method, you will always end up with two identical factors when you complete Step #3.

Now that we know that the factors of x² + 12x +36 are equal to (x+6)², we can solve for x as follows:

  • x² + 12x +36 = 4

  • (x+6)² = 4

  • √[(x+6)²] = √[4]

  • x + 6 = ± 2

In this case, the original quadratic function x² +12x +32 = 0 will have two solutions:

  • x + 6 = 2

  • x + 6 = -2

We can determine these two solutions by solving each equation as follows:

  • x + 6 = 2 → x = -4

  • x + 6 = -2 → x = -8

The entire 3-step method for completing the square for Example #2 is shown in Figure 05 above.

Final Answer: x = -4 and x = -8

Figure 06 below shows the graph of the parabola represented by x² +12x +32, with x-intercepts at -4 and -8.

 

Figure 06: Solving a quadratic by completing the square helps you to find the x-intercepts of the parabola that represents the equation.

 

How to Complete the Square: Example #3

Solve: x² +2x -7 = 0

Are you starting to get the hang of how to complete the square? Let’s gain some more experience with this next example.

Step #1: Rearrange the quadratic equation so that all of the constants are on one side of the equals sign.

For this problem, we know that a=1, b=2, and c=-7.

For our first step, let’s rearrange the equation so that all of the constants are on the right side:

  • x² +2x -7 = 0

  • x² +2x -7 (+7) = 0 (+7)

  • x² + 2x = 7

Now we have a new equivalent function:

  • x² + 2x = 7

Step #2: Add (b/2)² to both sides of the equal sign.

Now we need to add (b/2)² to both sides of the equation. Since b=2 in this example, (b/2)² is equal to:

  • (b/2)²

  • (2/2)²

  • (1)²

  • =1

Since (b/2)² = 1, we can complete the second step by adding 1 to each side of the equation as follows:

  • x² + 2x = 7

  • x² + 2x +1 = 7 +1

  • x² + 2x +1 = 8

 

Figure 07: How to Complete the Square: Solutions will not always be rational numbers.

 

Step #3: Factor and solve.

Now we are ready to factor and solve the equation.

We have a trinomial on the left side of the equation that can be factored as follows:

  • x² + 2x +1 = (x+1)(x+1)

  • x² + 2x +1 = (x+1)²

With these factors in mind, we can solve for x as follows:

  • x² + 2x +1 = 8

  • (x+1)² = 8

  • √[(x+1)²] = √[8]

  • x +1 = ± √[8]

For this third example, the quadratic function x² +2x -7 = 0 will have two solutions:

  • x + 1 = √[8]

  • x + 1 = - √[8]

If we continue onto solving these two equations, we will see that, unlike Examples #1 and #2, we do not end up with a rational answer:

  • x + 1 = √[8] → x = -1 + √8

  • x + 1 = - √[8] → x = -1 - √8

In cases like this, you can often conclude that:

Final Answer: x = -1 + √8 and x = -1 - √8

The solutions above are considered exact answers. However, if you are trying to estimate where the parabola will cross the x-axis on the coordinate plane, you could take the problem a step further by approximating for √8 as follows:

  • √8 ≈ 2.83

  • x = -1 + √8 → x = -1 + 2.83 → x =1.83

  • x = -1 + √8 → x = -1 - 2.83 → x =-3.83

Now we have two approximate solutions for x:

  • x =1.83 and x=-3.83

What does this mean? Just like we saw in Examples #1 and #2, the solutions tell you where the graph of the parabola crosses the x-axis. In this example, the graph crosses the x-axis at approximately 1.83 and -3.83, as shown in Figure 08 below.

 

Figure 08:  How to complete the square to determine the solutions to a quadratic equation.

 

Conclusion: How to Complete the Square

When learning how to solve quadratic equations of the form ax² + bx +c=0, understanding how to complete the square to find the values where x=0 is an important and useful algebra skill that you can use to solve a variety of problems.

Whenever you have an equation in ax² + bx +c=0 form, you can solve it by following these 3-simple steps to completing the square:

  • Step #1: Rearrange the quadratic equation so that all of the constants are on one side of the equals sign.

  • Step #2: Add (b/2)² to both sides of the equal sign.

  • Step #3: Factor and solve.

Note that the above 3-step method for completing the square can be used to find the solutions of any quadratic equation of the form ax² + bx +c=0. These solutions represent the x-values where the parabola that represents the equation crosses the x-axis.

Keep Learning:

How to Find the Vertex of a Parabola in 3 Easy Steps

Learn how to find the coordinates of the vertex point of any parabola with this free step-by-step guide.

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How to Find Horizontal Asymptotes in 3 Easy Steps

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How to Find Horizontal Asymptotes in 3 Easy Steps

How to Find Horizontal Asymptotes in 3 Easy Steps

Step-by-Step Guide: How to Find the Horizontal Asymptote of a Function

 

Step-by-Step Guide: How to Find the Horizontal Asymptotes of Functions.

 

During your study of algebra, you will eventually learn how to analyze and understand the behavior functions when they are represented graphically on the coordinate plane. The graph of a function gives you a visual portrayal of how the function behaves—including the location of any potential horizontal asymptotes.

This free step-by-step guide on How to Find Horizontal Asymptotes of a Function will demonstrate and explain everything you need to know about horizontal asymptotes and how to find the horizontal asymptote of any function (assuming that the function does indeed have a horizontal asymptote).

The following sections will be covered in this guide. While we recommend that you work through each section in order, you can use the quick-links below to jump to a particular section":

Let’s start off by learning more about horizontal asymptotes and what they look like.

 

Figure 01: The horizontal asymptote of a function tells you the behavior of a function as it approaches the edges of a graph.

 

What is a Horizontal Asymptote?

Before you learn how to find horizontal asymptotes, it is important for you to understand some key foundational concepts.

In algebra, a function is an equation of the form y=f(x) that represents the relationship between two sets. A relation can be defined as a function when there is only one output value for each input value (i.e. the x-values do not repeat). Functions can be visually represented on a graph (i.e. a coordinate plane).

A horizontal asymptote of a function is a horizontal line (↔) that tells you the behavior of a function as it approaches the edges of a graph. We can also say that horizontal asymptotes allow us to identify the “end behavior” of a function.

It is also important to note that horizontal asymptotes occur when functions are rational expressions, meaning that the function is a quotient of two polynomials (i.e. the function is a fraction where both the numerator and the denominator are polynomials.

For example, consider the following function:

  • y = (3x²)/x³

 

Figure 02: The graph of the function y = (3x²)/x³

 

Our function, y = (3x²)/x³, is the quotient of two polynomials (i.e. it is a rational expression) and its corresponding graph is shown in Figure 02 above.

Since our function is a fraction, we know that we can not have zero in the denominator (otherwise the function would be undefined at this point). So, the graph of the function, as you can see in Figure 02, does something interesting at zero.

In this example, we have a horizontal asymptote at y=0.

Notice how the graph of the function y = (3x²)/x³ gets closer and closer to the line y=0, without ever touching it, as it approaches the ends of the graph (horizontally), as shown in Figure 03 below.

 

Figure 03: The function has a horizontal asymptote at y=0.

 

If we take a look at a few points on the graph of y = (3x²)/x³ and the corresponding table, as shown in Figure 04 below, we can see that as the graph moves to the left of the coordinate plane, the y-values get closer and closer to zero. And, we can also see that as the graph moves to the right of the coordinate plane, the y-values get closer and closer to zero.

This type of observation is what we call determining the “end behavior” and it helps us to understand why the graph has a horizontal asymptote at y=0.

 

Figure 04: How to Find a Horizontal Asymptote of a Function

 

Now that you understand what a horizontal asymptote is and what it looks like, you are ready to work through a few step-by-step examples of how to find a horizontal asymptote.

Before moving forward, here are a few quick key points you should be familiar with:

  • Not all functions that are rational expressions have a horizontal asymptote.

  • It is possible for a function to have 0, 1, or 2 asymptotes (i.e. a function can have a maximum of 2 asymptotes).

  • When horizontal asymptotes are shown on a graph, they are typically drawn using a dashed line, which is what you will see in this guide.

  • The horizontal asymptote of a function is not a part of the function, and it is not a requirement to include the horizontal asymptote of a function when you graph it on the coordinate plane.

  • A horizontal asymptote can be thought of as an imaginary dashed line on the coordinate plane that helps you to visual a “gap” in a graph.

 

Figure 05: A horizontal asymptote is an imaginary line that is not a part of the graph of a function.

 

How to Find Horizontal Asymptotes Example #1

Now you are ready to learn how to find a horizontal asymptote using the following three steps:

  • Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

  • Step Two: Determine lim x→-∞ f(x). In other words, find the limit for the function as x approaches negative ∞.

  • Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Now, let’s go ahead and apply these three steps to our first practice problem.

Example #1: Find the horizontal asymptote of the function f(x)=4x/(x+4)

For this first example, we can start by taking a look at the graph in Figure 05 above. By looking at the graph, we can see that the function f(x)=4x/(x+4) has one horizontal asymptote at y=4.

Let’s now apply our three steps to see if confirm the at the function has a horizontal asymptote at y=4.

Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

  • lim x→∞ f(x) = lim x→∞ 4x/(x+4)

  • = 4 lim x→∞ [1/(1+(4/x))]

  • = 4 (lim x→∞ (1)) / (lim x→∞ (1+(4/x))

  • = 4 ( (1)/(1))

  • =4

After completing the first step, we can see that, as the function has a horizontal asymptote at y=4 as the function approaches positive ∞ on the graph.

Next, let’s confirm that the function has the same behavior as it approaches negative ∞.

Step Two: Determine lim x→-∞ f(x). In other words, find the limit for the function as x approaches negative ∞.

  • lim x→-∞ f(x) = lim x→∞ 4x/(x+4)

  • = 4 lim x→-∞ [1/(1+(4/x))]

  • = 4 (lim x→-∞ (1)) / (lim x→∞ (1+(4/x))

  • = 4 ( (1)/(1))

  • =4

Completing the second step gives us the same result as the first step (i.e. there is a horizontal asymptote at y=4).

Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Finally, based on our results from steps one and two, we can conclude that the function has one horizontal asymptote at y=4, as shown on the graph in Figure 06 below.

Final Answer: The function has one horizontal asymptote at y=4.

 

Figure 06: How to find a horizontal asymptote using limits.

 

How to Find Horizontal Asymptotes Example #2

Find the horizontal asymptote of the function f(x)=3ˣ+5

For this next example, we want to see if the exponential function f(x)=3ˣ+5 has any horizontal asymptotes.

We can solve this problem the same as we did the first example by using our three steps as follows:

Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

  • lim x→∞ f(x) = lim x→∞ 3ˣ+5

  • =lim x→∞ (3ˣ) + lim x→∞ (5)

  • lim x→∞ (3ˣ) = ∞

  • lim x→∞ (5) = 5

  • = ∞ + 5 = ∞

Notice that our result for the first step is that, as the function approaches positive ∞, the limit is ∞. And, since ∞ is not a real number, we can not yet determine whether or not this function will have any horizontal asymptotes. But we can not be completely sure until we complete the second step.

Step Two: Determine lim x→-∞ f(x. In other words, find the limit for the function as x approaches negative ∞.

  • lim x→-∞ f(x) = lim x→-∞ 3ˣ+5

  • =lim x→-∞ (3ˣ) + lim x→-∞ (5)

  • lim x→-∞ (3ˣ) = 0

  • lim x→-∞ (5) = 5

  • = 0 + 5 = 5

Our result from step number two results in a real number, so we can conclude that the function does have a horizontal asymptote at y=5.

Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Again, even though Step One did not result in a real number limit, Step Two did, so we can conclude that:

Final Answer: The function has one horizontal asymptote at y=5.

We can confirm the location of a horizontal asymptote at y=5 for the exponential function f(x)=3ˣ+5 by looking at the completed graph in Figure 07 below.

 

Figure 07: How to find horizontal asymptotes given a function explained.

 

How to Find Horizontal Asymptotes Example #3

Find the horizontal asymptote of the function f(x)=(3x^2+x)/(x+2)

For this third and final example, we have to see if the rational function f(x)=(3x^2+x)/(x+2) has one, two, or zero horizontal asymptotes.

We can determine whether or not the function has any horizontal asymptotes by following our three steps as follows:

Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

  • lim x→∞ f(x) = lim x→∞ (3x^2+x)/(x+2)

  • =lim x→∞ (3x+1)/(1+(2/x))

  • = (lim x→∞ (3x+1)) / (lim x→∞(1+(2/x)))

  • lim x→∞ (3x+1) = ∞

  • lim x→∞(1+(2/x)) = 1

  • = ∞ / 1 = ∞

Since our result is not a real number, Step One does not help us to determine the possible existence of a horizontal asymptote.

Step Two: Determine lim x→-∞ f(x. In other words, find the limit for the function as x approaches negative ∞.

  • lim x→-∞ f(x) = lim x→-∞ (3x^2+x)/(x+2)

  • =lim x→-∞ (3x+1)/(1+(2/x))

  • = (lim x→-∞ (3x+1)) / (lim x→-∞(1+(2/x)))

  • lim x→-∞ (3x+1) = -∞

  • lim x→-∞(1+(2/x)) = 1

  • = -∞ / 1 = -∞

Again, our result is not a real number so we can not determine the location of a horizontal asymptote.

Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Since neither Step One nor Step Two resulted in a real number value for k, we can conclude that the function does not have any horizontal asymptotes.

Final Answer: The function does not have any horizontal asymptotes.

The graph of the function f(x)=(3x^2+x)/(x+2) is shown in Figure 08 below. Do you notice how there is no horizontal asymptote?

 

Figure 08: How to Find Horizontal Asymptotes: Not all functions will have a horizontal asymptote.

 

Conclusion: How to Find Horizontal Asymptotes

A horizontal asymptote of a function is an imaginary horizontal line (↔) that helps you to identify the “end behavior” of the function as it approaches the edges of a graph.

Not every function has a horizontal asymptote. Functions can have 0, 1, or 2 horizontal asymptotes.

If a function does have any horizontal asymptotes, they will be displayed as a dashed line. A horizontal asymptote is an imaginary line that is not a part of the function, and it is not a requirement to include the horizontal asymptote of a function when you graph it on the coordinate plane.

You can determine whether or not any function has horizontal asymptotes by following these three simple steps:

  • Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

  • Step Two: Determine lim x→-∞ f(x). In other words, find the limit for the function as x approaches negative ∞.

  • Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Keep Learning:

How to Find the Vertex of a Parabola in 3 Easy Steps

Learn how to find the coordinates of the vertex point of any parabola with this free step-by-step guide.

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