How to Simplify a Square Root in 3 Easy Steps

How to Simplify a Square Root in 3 Easy Steps

How to Simplify a Square Root

Step-by-Step Guide: How to Simplify a Square Root in 3 Steps

 

Free Step-by-Step Guide: Ready to learn how to simplify a square root?

 

Simplifying square roots is a useful and important math skill that every student can learn with enough practice. By learning a simple 3-step process for simplifying square roots, you can learn to quickly and correctly simplify any square root (whether it is a perfect square or not), and that is exactly what we will be doing in this free guide.

The sections below will teach you exactly how to simplify a square root using a simple step-by-step method. Together, we will recap some key concepts and vocabulary terms and then work through three examples of how to simplify a square root. Whether you are learning this skill for the very first time or you are an experienced student in need of a quick and comprehensive review, this page will share everything you need to know about how to simplify a square root.

This guide is organized based on the following sections:

You can use the text links above to jump to any section of this guide, or you can work through the sections in order. Let’s get started!

 

Preview: How to Simplify a Square Root in 3 Steps.

 

Quick Intro: Square Roots and Perfect Squares

Before we work through examples of how to simplify a square root, let’s quickly recap some important concepts and vocabulary terms related to this topic.

In math, square roots are the inverse (or opposite) operation of squaring a number (i.e. multiplying a number by itself). And, conversely, the square root of a number is the value that, when multiplied by itself, results in the number that you started with.

For example, consider the square root of 16, which can be expressed using square root notation:

  • √16

We can say that √16 equals 4 because 4 times itself (i.e. 4x4 or 4²) equals 16, therefore:

  • √16 = 4 → because 4² = 16

Numbers like 16 are called perfect squares because their square roots are whole numbers, which makes them very easy to simplify.

In fact, you are likely already familiar with how to simplify many perfect squares such as:

  • √4 = 2 → because 2² = 4

  • √9 = 3 → because 3² = 9

  • √16 = 4 → because 4² = 16

  • √25 = 5 → because 5² = 25

  • √36 = 6 → because 6² = 36

  • √49 = 7 → because 7² = 49

  • √64 = 8 → because 8² = 64

  • √81 = 9 → because 9² = 81

  • √100 = 10 → because 10² = 100

Figure 01: Perfect Squares up to 144

Make sure that you know all of the perfect squares from 1 to 144 (as shown in Figure 01 above) before moving onto the next section.

While it is relatively easy to simplify a square root of a perfect square, it is more complicated to simplify a square root of a non-perfect square, which is what we will be focusing on in the next section, where, for example, we will look at finding the perfect squares of numbers like 18, 75, and 112.

How to Simplify a Square Root Examples

Now we are ready to use the following simple 3-step method for simplifying square roots to solve three practice problems:

Steps: How to Simplify a Square Root

  • Step 1: Identify two factors where one of them is a perfect square (choose the largest perfect square factor), and rewrite as a product.

  • Step 2: Split the product using two square root symbols.

  • Step 3: Simplify the perfect square and rewrite your final answer.

Example A: Simplify √18

 

Example A: Simplify √18

In our first example, we want to simplify a square root of a non-perfect square: √18.

We can simplify a square root like √18 by using our three step strategy as follows:

Step 1: Identify two factors where one of them is a perfect square.

Let’s start by listing the factors of 18:

  • Factors of 18: 1, 2, 3, 6, 9 and 18

Notice that 18 has one factor that is a perfect square: 9, and that:

  • 9 x 2 = 18

Since 9 x 2 equals 18, the two numbers that we are going to use for Step 2 are 9 and 2.

Step 2: Split the product using two square root symbols.

For the second step, we can use the factors from Step 1 to rewrite √18 as follows:

  • √18 = √(9 x 2) = √9 x √2

We can “split” the square root in this way because of the product property of square roots, which says that:

  • √(A x B) = √(A) x √(B)

(provided that A and B are non-negative numbers).

So we now have a new equivalent product that represents √18, which is…

  • √9 x √2

Step 3: Simplify the perfect square and rewrite your final answer.

For our final step, notice that one of the square roots in our new expression, √9, is a perfect square. Since we know that √9 = 3, we can rewrite √9 as 3 as follows:

  • √9 x √2 = 3 x √2

Now, all that we have to do is rewrite the result, 3 x √2, as 3√2 , and we have solved the problem!

Final Answer: √18 = 3√2

Figure 02 below shows the step-by-step process for simplifying this square root.

 

Figure 02: How to Simplify a Square Root in 3 Steps.

 

Now that you have learned how to simplify a square root, let’s gain some more experience by working through another example.

Example B: Simplify √75

 

Example B: Simplify √75

We can solve this next example by using the three steps that we used to solve the previous example.

Step 1: Identify the factors of 75 and determine the largest perfect square factor.

We can begin by listing the factors of 75:

  • Factors of 75: 1, 3, 5, 15, 25 and 75

Notice that 75 has one perfect square factor, 25, and that:

  • 25 x 3 = 75

Step 2: Split the product using two square root symbols.

Next, we can use the factors from Step 1 to rewrite √75as follows:

  • √75 = √(25 x 3) = √25 x √3

Step 3: Simplify and solve.

Finally, we can simplify √25 as 5 (since √25=5) and rewrite the expression as follows:

  • √25 x √3 = 5 x √3

We can now rewrite 5 x √3 as 5√3 and we can conclude that:

Final Answer: √75 = 5√3

The entire process for solving this problem is shown in Figure 03 below.

 

Figure 03: Multiplying fractions with whole numbers Example B solved.

 

Example C: Simplify √112

 

Example C: Simplify √112

For our final step-by-step example of how to simplify a square root, let’s take on a triple-digit number using our three-step method.

Step 1: Identify the factors of 112 and pick out the largest perfect square factor.

  • Factors of 112: 1, 2, 4, 7, 8, 14, 16, 28, 56 and 112

Notice 112 has two perfect square factors: 4 and 16. In cases like this, always choose the largest perfect square factor (16 in this case).

Now that we have identified our perfect square factor, we can say that:

  • w16 x 7 = 112

Step 2: Split the product using two square root symbols.

Next, we can write √112 as follows:

  • √112 = √(16 x 7) = √16 x √7

Step 3: Simplify and solve.

For the final step, we can simplify √16 as 4 and rewrite the expression as follows:

  • √16 x √7 = 4 x √7

Now we just have to rewrite 4 x √7 as 4√7 and we have solved the problem!

Final Answer: √112 = 4√7

The three-step process for solving Example C is shown in Figure 03 below

 

Figure 03: How to Simplify a Square Root in 3 Steps.

 
 
 

More Free Math Resources:

How to Multiply Fractions with Whole Numbers—Step-by-Step

How to Multiply Fractions with Whole Numbers—Step-by-Step

How to Multiply Fractions with Whole Numbers

Step-by-Step Guide: How to Multiply Fractions with Whole Numbers, Multiplying Fractions by Whole Numbers Examples

 

Free Step-by-Step Guide: How to multiply fractions with whole numbers explained.

 

Multiplying fractions with whole numbers can seem like a challenging math skill, but, with some simple strategies and an easy step-by-step method, it can be a relatively easy task that any student can master.

In this free guide, we will work through several examples of how to multiply fractions with whole numbers using a simple step-by-step process. As long as you can follow three easy steps, you will be able to confidently and accurately solve a variety of math problems where you have to multiply fractions with whole numbers.

You can work through the sections in this free guide in sequential order, or you can click on any of the quick-links below to jump to one particular section.

Quick Intro: Multiply Fractions with Whole Numbers

Before we dive into any examples of how to multiply fractions with whole numbers, let’s do a quick introductory review of what it means when we multiply fractions with whole numbers.

For example, let’s consider the example 3 x 1/4:

  • 3 is the whole number

  • 1/4 is the fraction

Whenever you multiply a fraction by a whole number, you are really just performing repeated addition (i.e. you are adding the fraction to itself a number of repeated times that is determined by the whole number).

If we think of multiplication in terms of repeated addition, we can rewrite 3 x 1/4 as follows:

  • 3 x 1/4 = 1/4 + 1/4 + 1/4

And, since 1/4 + 1/4 + 1/4 is equal to 3/4, we can conclude that:

  • 3 x 1/4 = 1/4 + 1/4 + 1/4 = 3/4

Final Answer: 3 x 1/4 = 3/4

 

Figure 01: How to Multiply Fractions with Whole Numbers Using Repeated Addition.

 

The process of multiplying fractions with whole numbers using repeated addition is shown in Figure 01.

While we will not use repeated addition to solve the examples in this guide, understanding this basic relationship between multiplication and repeated addition is the first step to easily learning how to multiply fractions with whole numbers.

Now, let’s go ahead and work through some examples of multiplying fractions with whole numbers using a simple 3-step method.

Multiplying Fractions by Whole Numbers Examples

For all of the multiplying fractions with whole numbers examples that follow, we will be using the following 3-step method for solving:

  • Step 1: Rewrite the whole number as a fraction with a denominator of 1.

  • Step 2: Multiply the numerators together and then multiply the denominators together.

  • Step 3: Simplify if possible.

 

Example A: Multiplying whole numbers with fractions.

 

Example A: Multiply 2 x 1/3

For our first example, we have to multiply the whole number 2 by the fraction ⅓, and we will do that by following our 3-step process as follows:

Step 1: Rewrite the whole number as a fraction with a denominator of 1.

First, we can rewrite the whole number, 2, as a fraction with a numerator of 1 as follows:

  • 2 → 2/1

Now we have a new multiplication problem:

  • 2 x 1/3 → 2/1 x 1/3

Step 2: Multiply the numerators together and then multiply the denominators together.

Now that we have a new expression with two fractions being multiplied by each other, we can perform multiplication by multiplying the numerators together and then multiplying the denominators together as follows:

  • 2/1 x 1/3 = (2x1) / (1x3) = 2/3

After completing Step 2, we are left with the fraction 2/3.

Step 3: Simplify if possible.

Finally, we just have to check if our result from Step 2, 2/3, can be simplified.

In this case, the fraction 2/3 can not be simplified because there is no common factor between the numerator (2) and the denominator (3) other than 1.

Final Answer: 2 x 1/3 = 2/3
The complete step-by-step process for solving this first example is shown in Figure 02 below.

 

Figure 02: How to multiply fractions with whole numbers step-by-step.

 

Now that you are familiar with our 3-step method for multiplying fractions with whole numbers, let’s gain some more experience by using them to solve another example.

 

Example B: 2/3 × 34 = ?

 

Example B: Multiply 2/3 x 4

For this next example, notice how, in this case, the first term is a fraction and the second term is the whole number (this is a reverse situation compared to Example A). However, the commutative property of multiplication tells us that the order of the terms does not matter, so we can still use our 3-step process to solve this problem as follows:

Step 1: Rewrite the whole number as a fraction with a denominator of 1.

We can leave the fraction 2/3 alone and rewrite the whole number 4 as a fraction with a denominator of 1 as follows:

  • 2/3 x 4 → 2/3 x 4/1

Step 2: Multiply the numerators together and then multiply the denominators together.

Next, we can take our new expression and simply multiply the numerators together, and then the denominators together as follows:

  • 2/3 x 4/1 = (2x4) / (3x1)

  • (2x4) / (3x1) = 8/3

After completing the second step, our result is 8/3. We can now move onto the third and final step.

Step 3: Simplify if possible.

Let’s see if our result from Step 2, 8/3, can be simplified.

Since there is no common factor (other than 1) between the numerator (8) and the denominator (3), we know that the fraction 8/3 can not be simplified. However, since 8/3 is an improper fraction, we do have the option of either expressing it as 8/3 or as the mixed number 2 2/3 (in this case, we will choose to express our answer as 8/3).

Final Answer: 2/3 x 4 = 8/3
Figure 03 below illustrates our step-by-step process for solving this second example.

 

Figure 03: Multiplying fractions with whole numbers Example B solved.

 
 

5Example C: 10 × 1/5 = ?

 

Example C: Multiply 10 x 1/5

Let’s gain some more practice using our 3-step method for multiplying fractions with whole numbers.

Step 1: Rewrite the whole number as a fraction with a denominator of 1.

In this example, we have to rewrite the whole number (10) as a fraction with a denominator of 1.

  • 10 x 1/5 → 10/1 x 1/5

Step 2: Multiply the numerators together and then multiply the denominators together.

Next, let’s take our new expression, 10/1 x 1/5, and multiply the numerators and denominators together:

  • 10/1 x 1/5 = (10x1) / (1x5)

  • (10x1) / (1x5) = 10/5

Finally, let’s move onto Step 3 to see if our result (10/5) can be simplified.

Step 3: Simplify if possible.

In this case, the numerator (10) and the denominator (5) share a common factor of 5. So, we can simplify 10/5 by dividing both the numerator and denominator by 5 as follows:

  • 10 ÷ 5 = 2

  • 5 ÷ 5 = 1

After dividing, we can say that 10/5 = 2/1, and we can rewrite 2/1 as 2.

Final Answer: 10 x 1/5 = 10/5 = 2/1 = 2
The entire process of solving Example C is shown in Figure 04 below illustrates our step-by-step process for solving this second example.

 

Figure 05: Example C Solved.

 
 

Example D: 7 × 5/6 = ?

 

Example D: Multiply 7 x 5/6

Let’s work through one final example of multiplying fractions with whole numbers.

Step 1: Rewrite the whole number as a fraction with a denominator of 1.

In this case, we can rewrite the whole number (7) as a fraction as follows:

  • 7 x 5/6 → 7/1 x 5/6

Step 2: Multiply the numerators together and then multiply the denominators together.

Next, we can multiply the two fractions together as follows:

  • 7/1 x 5/6 = (7x5) / (1x6)

  • (7x5) / (1x6) = 35/6

Step 3: Simplify if possible.

Finally, we have to see if our result from Step 2 (35/6) can be simplified. Since 35 and 6 do not share any common factors besides 1, we know that it can not be simplified any further.

Final Answer: 7 x 5/6 = 35/6
Figure 06 shows how we solved this final example.

 

Figure 06: How to Multiply Fractions with Whole Numbers.

 
 
 

More Free Math Resources:

3rd Grade Word Problems—Free PDF Worksheet Library

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3rd Grade Word Problems—Free PDF Worksheet Library

3rd Grade Word Problems—Free PDF Worksheets with Answer Keys

Looking for Free Printable 3rd Grade Math Worksheets?

 

Are you looking for free and engaging 3rd grade math word problems worksheets to share with your students?

 

Whether you are a 3rd grade classroom teacher or a parent of a 3rd grade student, you could use some free and engaging word problems for 3rd grade students to help them to develop important foundational math skills. This page shares a huge collection of 3rd grade word problems that cover topics including addition, subtraction, two-step problems, elapsed time, and more.

Jump to a Topic:

Whether you’re in need of worksheets for addition or subtraction word problems, two-step word problems for 3rd grade students, telling time and elapsed time word problems, area and perimeter word problems, or even measurement word problems, the 3rd grade word problems worksheet collection below will surely have something for you.

 

All of our 3rd grade math word problems worksheets are easy to print and share in your classroom.

 

3rd Grade Word Problems: Single-Digit Addition

▶ Math Skill Focus: Simple Addition, Adding Single-Digit Numbers

The following 3rd Grade Word Problems focus on basic addition in real-world scenarios.

You can preview any of the worksheets in this collection by clicking on any of the image boxes below, and you can download the corresponding PDF file by clicking on any of the blue text links. Each PDF file will include a set of word problems for 3rd grade students followed by a complete answer key on the last page.

For example, Dotty made 2 sugar cookies and 7 chocolate chip cookies. How many cookies did she make in total?

Worksheet A

Worksheet B

Worksheet C

These 3rd grade word problems worksheets focus on the basic foundational skill of adding single-digit numbers in a real-world context. They require students to identify key information, use mathematical thinking, correctly perform simple addition, and express their answer in writing.

3rd Grade Word Problems: Double-Digit Addition

▶ Math Skill Focus: Simple Addition, Adding Double-Digit Numbers

Once your students have mastered solving single-digit word problems for 3rd grade, the next step is to work through similar problems that involve adding two-digit numbers to solve word problems related to real-world scenarios.

For example, Elly is making donuts to sell at a local bake sale. He bakes 24 chocolate donuts, 21 vanilla donuts, and 15 cinnamon donuts. How many donuts did Elly bake?

Worksheet A

Worksheet B

Worksheet C

Do you want more free 3rd Grade Math Activities in your inbox every week?

 
 

3rd Grade Word Problems: Single-Digit Subtraction

▶ Math Skill Focus: Simple Subtraction, Subtracting Single-Digit Numbers

The following 3rd Grade Word Problems focus on basic subtraction in real-world scenarios. Once students have mastered these types of 3rd grade math word problems, they can move onto the double-digit subtraction word problems in the next section.

For example, Ruben planted 12 flower seeds in his garden. After the first week, 3 of the seeds sprouted. After the second week, 5 more of the seeds sprouted. How may of the seeds did not sprout?

Worksheet A

Worksheet B

Worksheet C

3rd Grade Word Problems: Double-Digit Subtraction

▶ Math Skill Focus: Simple Subtraction, Subtracting Double-Digit Numbers

Once your students are comfortable with solving 3rd grade math word problems involving single-digit subtraction, they can take the next step to solving problems involving finding the difference of two-digit numbers in a word problem format.

For example, Bethany has to cleanup after a dinner party. She has to wash 28 dishes in total. She has already washed 12 of the dishes. How many dishes does she have left to wash?

Worksheet A

Worksheet B

Worksheet C

In the next section, we will share some word problems for 3rd grade that focus on mixed addition and subtraction, where students will have to extend their thinking to using the context clues from each problem to determine whether they have to perform addition or subtraction to solve each problem.

3rd Grade Word Problems: Mixed Addition and Subtraction

▶ Math Skill Focus: Mixed Addition and Subtraction, Word Problem Solving

This next set of 3rd grade math word problems require students to use their math skills to determine whether or not they have to use addition or subtraction to solve each word problem.

For example, There were 24 notebooks in a bin. Students took 11 for their backpacks. How many notebooks are left in the bin?

3rd Grade Word Problems: Two-Step Word Problems

▶ Math Skill Focus: Use addition and/or subtraction to solve two-step word problems

One of the biggest differences between 2nd grade math and 3rd grade math is that 3rd graders begin learning how to solve word problems that require multiple steps to solve.

For example, Joey has 10 apples. He gives 4 apples to Josh and 3 apples to Jane. How many apples does Joey have left?

Worksheet A

Worksheet B

Worksheet C

The next two sections will share word problems for 3rd grade students that focus on two auxiliary topics: elapsed time and area and perimeter of rectangles.

3rd Grade Word Problems: Elapsed Time

▶ Math Skill Focus: Solve word problems involving time, units of time (minutes, hours, etc.), and elapsed time

Outside of learning how to solve word problems involving operations, another important 3rd grade math topic is dealing with time, units of time, and elapsed time. This section shares three 3rd grade word problems worksheets related to elapsed time.

For example, Jackson started his homework at 3:15 PM. He finished at 3:50 PM. How many minutes did he spend on completing his homework?

Worksheet A

Worksheet B

Worksheet C

3rd Grade Word Problems: Area and Perimeter of Rectangles

▶ Math Skill Focus: Solve word problems involving area and perimeter of rectangles

This final section includes 3rd grade math word problems worksheets on finding the area and/or perimeter of rectangular figures in real-world scenarios.

For example, Aaron is building a small rectangular flower box that is 5 feet long and 3 feet wide. What is the area, in square feet, of Aaron’s flower box?

Note that Worksheet A focuses on area only, Worksheet focuses on perimeter only, and Worksheet C involved mixed area and perimeter 3rd grade math word problems.

Worksheet A

Worksheet B

Worksheet C

Looking for More 3rd Grade Practice Worksheets?

Be sure to visit our Free 3rd Grade Math Worksheet Library, which shares hundreds of free PDF practice worksheets for a variety of 3rd grade topics.

Helpful Hints for Solving 3rd Grade Math Word Problems

The focus of our 3rd grade word problems worksheets is to give early elementary students to apply their procedural math skills to real-world situations and scenarios. Rather than just asking students to solve a simple math operation problem (e.g. 9 + 7 = ?), these types of problems are more advanced, as they require students to apply reading comprehension and to use context clues to find answers.

Since math word problems are more advanced, many students will initially struggle with them, and it takes time and practice to get better at solving 3rd grade word problems (one-step or two-step) whether they involve addition, subtraction, multiplication, mixed operations, elapsed time, or area and perimeter.

If your 3rd grader is having a hard time with solving any of the problems on any of our 3rd grade word problems worksheets, here are a few helpful hints to improve their chances of correctly solving any given word problem:

  • Read Each Question Carefully: One of the biggest reasons why 3rd graders struggle with word problems is because they fail to read each question carefully and correctly assess exactly what the problem is asking them to do. Before students attempt to solve a math word problem, they need to read to problem, identify and keywords or important information, and identify exactly what the question is asking them to do. Many students will benefit greatly from using a marker to underline key information or by using a colored highlighter.

  • Ask Questions Before You Start: While this advice applies to solving any math problem, it is particularly useful whenever students are working on math word problems. Once you have carefully read a question and identify the important information, you should ask yourself “what is this question asking me to do?” and “what will the final answer look like?”. These two questions help students come up with a plan before they attempt to solve a problem. For example, if a question asks students to find the area of a rectangle, they should be aware that their final answer will have to be in terms of square units.

Helpful Hints for Solving 3rd Grade Word Problems: Always show your work and answer using complete sentences.

  • Show Your Work: All of our 3rd grade word problems worksheets require students to show their work. But, what does showing your work actually look like? In addition to students writing out how they performed their operations, they should also be encouraged to use additional visual aids such as drawing diagrams or using tally marks. For example, when solving an area of a rectangle word problem, it is incredibly helpful to draw a rectangle and label the length of each side before attempting to solve the problem. Or, if a question involves combining a pile of 13 apples with a pile of 9 apples, students can draw each pile and then count the total number of apples.

  • Write Your Final Answer in Sentence Form: In math, word problems typically have “word answers”, so students should get used to expressing their final answer to any word problem by using a complete sentence. While this is a general rule, it is good practice for students. For example, when solving the problem “Ethan read 32 pages on Monday and 21 pages on Tuesday. He wants to read 120 pages this week. How many more pages does he need to read? “, the final answer is not just 67, but “Ethan needs to read 67 more pages.”

(Do you want free K-8 math resources and activities in your inbox every week? Click here to sign up for our free math education email newsletter)

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53 Funny Teacher Memes to Brighten Your Day!

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53 Funny Teacher Memes to Brighten Your Day!

53 Funny Teacher Memes Every Educator Can Relate To

Get Ready to Laugh Out Loud with These Funny Memes About Teaching!

Ready for Some Funny Teacher Memes?

Being a teacher is one of the toughest—and often most under-appreciated—careers a person can have.

And, while teaching can be an incredibly rewarding job with tons of positives (like making a difference in the daily lives of your students), there are also some negatives and frustrations that accompany the task of educating students for a living.

In the spirit of taking the bad with the good when it comes to be a teacher, it can fun be to celebrate the quirks of the profession, many of which only yourself and fellow teachers can truly understand.

So, let’s take a moment to celebrate the craft that is being a professional educator in a fun and humorous way by sharing the best and funniest teacher memes that the internet has to offer!

Below you will find our collection of the 53 best teacher memes, many of which are all too relatable to those who work inside of a classroom. Each teacher meme shines a light on some of the more frustrating and/or perplexing aspects of being a teacher at any grade level.

Whether you are dealing with rowdy children, out-of-touch administrators, delusional parents, or never-ending meetings and conferences that obviously could have been emails, this page surely has a meme or two that will have you rolling on the floor laughing in no time!

A hysterical teacher meme or two is just what you need to make it through your day!

Before you jump into our collection of 53 funny teacher memes, remember that each teach meme is meant to be playful and they are meant to be taken light-heartedly. Also remember that laughter is often the best medicine and it’s totally fine to laugh out loud at some of the vexing aspects of being an educator.

Whenever you are ready to laugh, scroll down to start enjoying our funny teacher memes and, if you find any memes to be exceptionally hysterical, then feel free to share this page with your fellow teachers to spread some much-needed joy and laughter :)

Funny Teacher Memes #1-10

1.) This can happen to the best of us.

 

Teacher Meme #1: When you forget to remember…

 

2.) I swear, I was only gone for two seconds!

 

Teacher Meme #2: 😬

 

3.) When the classroom telephone starts ringing the tenth time today…

 
 

4.) Yes, yessssss….

 

This may be the best teacher meme of all time!

 

5.) Your students doing anything to avoid actually doing their classwork…

 

Funny Teacher Meme via Reddit User u/aRabidGorilla

 

6.) All of the moisturizer in the world couldn’t save us!

 

Teaching Memes Every Educator Can Relate To!

 

7.) Is this even written in English?

 

Funny Teacher Memes #7

 

8.) Your reaction to accidentally overhearig your students’ personal conversations…

 
 

9.) You can not possibly be even remotely serious…

 

Teaching Memes #10

 

10.) When you’re in the middle of an awesome lesson and the fire alarm goes off…

 
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Funny Teacher Memes #11-20

Are you ready for the next ten hysterical teacher memes that will have you and your colleagues laughing out loud?

11.) We’re all thinking it…

 

Funny Memes About Teaching: What does the principle even do?

 

12.) When you start handing out rulers to your students…

 
 

13.) It’s sad—I mean, funny—because it’s true.

 

Teacher Memes: Sad, but True.

 

14.) May I laugh at this funny teacher meme?

 

Funny Teacher Memes #14

 

15.) You shall not pass!

 

Teacher Meme #11

 

16.) Good riddance!

 

Funny Teacher Memes #16

 

17.) When April Fools’ Day falls on the weekend….

 
 

18.) Dream on, dreamers…

 
 

19.) We actually just sat around and waited….

 

This teacher meme hits hard!

 

20.) If looks could kill…

 

53 Best Teacher Memes #20: Always show your work!

 

Funny Teacher Memes #21-30

Let’s keep the good times going with our next ten funny teacher memes!

21.) Every day is important!

 

Funny Teacher Memes #21

 

22.) Is anyone even listening to me?

 

My favorite teacher meme of all time!

 

23.) When you find out that you have to waste your off period sitting through another meeting…

 
 

24.) Just smile and nod…smile and nod…

 

Teacher Meme #24: Is this relatable to you?

 

25.) Some say that she’s still waiting…

 

Funny Teacher Memes

 

26.) This one is all too real!

 

Funny Memes About Teaching!

 

27.) Just gotta’ keep on truckin’…

 

This teacher meme is a little too real, isn’t it?

 

28.) When the copy machine is broken again…

 
 

29.) How sweet it is!

 

Funny Teacher Meme #29: Gotta savor the good stuff!

 

30.) When a student asks you how old you are…

 
 

Funny Teacher Memes #31-40

And now for ten more funny memes teacher edition.

31.) Let me just rest my eyes for a quick second…

 

Funny Teacher Memes: Teachers at home at 7:38pm…

 

32.) When a past student tells you that you were their favorite teacher…

 
 

33.) There can only be one…

 

Teacher Memes #33: The Teacher’s Dilemma.

 

34.) You couldn’t use the telephone and go on the internet at the same time!

 
 

35.) When a student’s math homework is written in pen…

 
 

36.) Wink, wink…

 

This is my favorite teacher meme of all time!

 

37.) Whatchu’ want me to say?

 

Teacher Meme #37: This one is spicy, we know!

 

38.) Back in my day….

 

Funny Teacher Memes #38

 

39.) Don’t pop the Champagne too early!

 

I have this teacher meme posted in my classroom!

 

40.) When you get an email telling you that your after-schooling meeting was cancelled…

 
 

Funny Teacher Memes #41-53

Our funny teacher memes collection wraps up with 13 more super funny teacher memes that will have your sides hurting!

41.) Feelin’ hot, hot, hot!

 

Can you relate to this funny teacher meme?

 

42.) Ya gotta’ love it!

 

Funny Teacher Memes #42

 

43.) When your class gets interrupted by a knock at the door…

 
 

44.) I guess it just doesn’t work that way…

 
 

45.) When you already explained something five different ways and students are still saying that they don’t get it…

 
 

46.) It takes two to Tango….

 

I also have this teacher meme posted in my classroom.

 

47.) When you find out that your students were well-behaved for the sub…

 
 

48.) If only a Panera Bread platter could solve all of our problems…

 

Is teacher meme #48 a truth bomb?

 

49.) No, I will not cover hall duty right now…

 
 

50.) The Parent-Teacher Conference Day experience in a nutshell…

 
 

51.) Most of us will probably go this way…

 
 

52) I’m not crying, you’re crying!

 

Funny Teacher Meme #52

 

53.) Every teacher after the last day of school before summer vacation…

 
 

That wraps up our list of the 53 funniest teacher memes that the internet has to offer! We hope that you these memes brightened up your day, and that you had a few laughs along the way. If you are looking for a few different ways that you can share and enjoy these teacher memes, here are a few ideas:

  • Revisit this page whenever the demands of teaching are weighing you down and you need a good chuckle or two to restore your sanity.

  • Share your favorite teacher memes in your teacher group chat or email chain to indulge in some humorous commiseration!

  • Post a funny teacher meme on social media to connect with fellow teachers, parents, and administrators.

  • Print and post a few funny teacher memes in your classroom to share some insight and humor with your students. This can be a fun way to give your students an idea of some of the challenges of being a teacher and working with students for a living.

  • Use these memes in meeting and/or professional development presentations to serve as an icebreaker or to simply break things up by injectiing some light-hearted and topical humor into the mix.

(Do you want more free K-8 math resources and activities in your inbox every week? Click here to sign up for our math education email newsletter)

Did you laugh, cry, or both? Share your reaction in the comments below!

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Quadratic Formula Examples—Solved Step-by-Step

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Quadratic Formula Examples—Solved Step-by-Step

Quadratic Formula Examples Tutorial

Step-by-Step Guide: Examples of How to Find the Roots of a Quadratic Function using the Quadratic Formula

 

Step-by-Step Guide: Quadratic Formula Examples Solved

 

Are you ready to work through a few quadratic formula examples to gain some more practice and experience with solving quadratic equations using the quadratic formula?

In math, the quadratic formula, x= (-b ± [√(b² - 4ac)]) / 2a is an incredibly important and useful formula that you can use to find the solutions (also known as roots) or any quadratic equation of the form ax² + bx + c = 0 (where a ≠ 0), whether it is easy to factor or not!

If you know how to use the quadratic formula, then you can solve a variety of algebra problems involving quadratic equations, and learning how to use it correctly is something that you can easily learn with some practice and repetition.

This free Quadratic Formula Examples Step-by-Step Guide includes a short review of the quadratic formula as well as several different practice problems that we will work through and solve using the quadratic formula with a step-by-step explanation. The guide is organize by the following sections, and you can click on any of the hyperlinks below to jump to any particular spot:

Before we dive into any of the quadratic formula examples, let’s start off with a quick review of the quadratic formula and why it is such a useful algebra tool.

 

Figure 01: The Quadratic Formula

 

What is the Quadratic Formula?

Before you can learn how to use the quadratic formula, it is important that you understand what a quadratic equation is.

Definition: A quadratic equation is a function of the form ax² + bx + c = 0 (where a does not equal zero). On a graph, a quadratic equation can be represented by a parabola. The x-values where the parabola crosses the x-axis is called the solutions, or roots, of the quadratic equation.

For example, consider the following quadratic equation:

  • x² + 5x + 6 = 0

Notice that this equation is in ax² + bx + c = 0 form, where…

  • a=1

  • b=5

  • c=6

If we want to find the solutions, or roots, of this quadratic equation, we have a few options.

First, we could factor this quadratic equation by looking for two values that add to 5 and also multiply to 6, which, in this case, would be 2 and 3. So we could say that…

  • x² + 5x + 6 = 0 → (x+2)(x+3) = 0

We could then solve for each factor as follows:

  • x + 2 = 0 → x = -2

  • x + 3 = 0 → x = -3

Now we can conclude that the solutions of this quadratic are x=-2 or x=-3.

 

Figure 02: What are the solutions (or roots) of a quadratic equation?

 

Another option for finding the solutions to a quadratic equation is to look at its graph. The solutions, or roots, will be the x-values where the graph crosses the x—axis. Note that quadratic equations can have two roots, one root, or even no real roots (as you will see later in this guide).

As for the equation x² + 5x + 6 = 0, the corresponding graph in Figure 02 above confirms that the equation has solutions at x=-2 and x=-3.

But what do we do when a quadratic equation is very difficult to factor or when we do not have access to a clear graph? Well, this is where the quadratic formula comes into play.

Definition: Any quadratic equation of the form ax² + bx + c = 0 (where a ≠ 0), can be solved using the quadratic formula, which states that…

  • x= (-b ± [√(b² - 4ac)]) / 2a

Why is the quadratic formula so useful? Because, as the definition states, it can be used to find the solutions to any quadratic equation. While the quadratic equation that we just looked at, x² + 5x + 6 = 0, was pretty easy to work with and solve, it is considered extremely simple. As you move farther along your algebra journey, you will come across more and more complex quadratic equations that can be very difficult to factor or even graph.

However, if you know how to use the quadratic formula, you can successfully solve any quadratic equation. With this in mind, let’s go ahead and work through some quadratic formula examples so you can gain some practice.

And we will start by using it to solve x² + 5x + 6 = 0, because we already know that the solutions are x=-2 and x=-3. If the quadratic formula works, then it should yield us that same result. Once we work through this first simple example, we will move onto more complex examples of how to use the quadratic formula to solve quadratic equations.

 

Figure 03: To use the quadratic formula, start by identifying the values of a, b, and c.

 

Quadratic Formula Examples

We will begin by using the quadratic formula to solve the equation shown in Figure 02 above: x² + 5x + 6 = 0

Example #1: Solve x² + 5x + 6 = 0

First, notice that our equation is in ax² + bx + c = 0 form where:

  • a=1

  • b=5

  • c=6

Identifying the values of a, b, and c will always be the first step (provided that the equation is already in ax² + bx + c = 0 form).

Now that we know the values of a, b, and c, we can plug them into the quadratic equation as follows:

  • x= (-b ± [√(b² - 4ac)]) / 2a

  • x= -(5) ± [√(5² - 4(1)(6))]) / 2(1)

  • x= -5 ± [√(25 - 24)] / 2

  • x= -5 ± [√(1)] / 2

  • x= (-5 ± 1) / 2

Now we are left with x= (-5 ± 1) / 2. Note that the ± mean “plus or minus” meaning that we have to split this result into two separate equations:

  • Plus: x = (-5 + 1) / 2

  • Minus: x = (-5 - 1) / 2

By solving these two separate equations, we can find the solutions to the quadratic function x² + 5x + 6 = 0.

  • x = (-5 + 1) / 2 = -4/2 = -2 → x=-2

  • x = (-5 - 1) / 2 = -6/2 = -3 → x=-3

After solving both equations, we are left with x=-2 and x=-3, which we already knew were the solutions to x² + 5x + 6 = 0. So, we have confirmed that the quadratic formula can be used to find the solutions to any quadratic equation of the form ax² + bx + c = 0.

Final Answer: x=-2 and x=-3

The steps to solving the quadratic formula example is illustrated in Figure 04 below.

 

Figure 04: Quadratic Formula Examples Step-by-Step

 

Example #2: Solve 2x² + 2x -12 = 0

For our next quadratic formula example, we will again start by identifying the values of a, b, and c as follows:

  • a=2

  • b=2

  • c=-12

Make sure that you correctly identify the sign (positive or negative) as well, since this is necessary to using to quadratic formula correctly.

Next, we can substitute these values for a, b, and c into the quadratic formula as follows:

  • x= (-b ± [√(b² - 4ac)]) / 2a

  • x= -(2) ± [√(2² - 4(2)(-12))]) / 2(2)

  • x= -2 ± [√(4 - -96)] / 4

  • x= -2 ± [√(100)] / 4

  • x= (-2 ± 10) / 4

Our result is x= (-2 ± 10) / 4. From here, we can rewrite e the result as two separate equations by “spitting” the ± sign as follows:

  • Plus: x= (-2 + 10) / 4

  • Minus:x= (-2 - 10) / 4

Now we can solve each individual equation to find the values of x that will be the solutions of this quadratic equation.

  • x= (-2 + 10) / 4 = 8/4 = 2 → x=2

  • x = (-2 - 10) / 4 = -12/4 = 3 → x=3

We are left with two values for x: x=2 and x=-3, and we can conclude that the quadratic equation 2x² + 2x -12 = 0 has the following solutions:

Final Answer: x=2 and x=-3

Figure 05 shows the step-by-step process for solving this quadratic formula example.

 

Figure 05: Quadratic Formula Examples #2 Solved

 

Example #3: Solve 2x² -5x + 3 = 0

For the next of our quadratic formula examples calls for us to use the quadratic formula to find the solutions to a quadratic function where:

  • a=2

  • b=-5

  • c=3

The process of substituting a, b, and c into quadratic formula will be exactly the same as the last two quadratic formula examples.

  • x= (-b ± [√(b² - 4ac)]) / 2a

  • x= -(-5) ± [√(-5² - 4(2)(3))]) / 2(2)

  • x= 5 ± [√(25 - 24)] / 4

  • x= 5 ± [√(1)] / 4

  • x= (5 ± 1) / 4

Are you starting to get the hang of it? Now that we have simplified our equation, we are left with x= (5 ± 1) / 4. And, just like the last two examples, we can go ahead and split this result into two separate equations as follows:

  • Plus: x= (5 + 1) / 4

  • Minus: x= (5 - 1) / 4

Finally, we just have to solve each equation to get our final answer (i.e. the values of the solutions).

  • x= (5+1) / 4 = 6/4 = 3/2 → x=3/2

  • x= (5-1) / 4 = 4/4 = 1 → x=1

Notice that the result of the first equation ended up as a fraction (3/2). This is totally fine! It just means that the parabola will cross the x-axis in the middle of a box (rather than hitting directly at an integer coordinate).

Final Answer: x=3/2 and x=1

All of the steps for solving this example are shown in Figure 06 below.

 

Figure 06: Sometimes a quadratic formula will give you a solution that is a fraction.

 

Example #4: Solve 3x² + 2 = 7x

The fourth and final of our quadratic formula examples looks a bit different. The given equation 3x² + 2 = 7x is not in ax² + bx + c = 0 form.

Whenever this is the case, we will have to see if we can use algebra to rearrange the equation so to make into ax² + bx + c = 0 form. We can do that by using inverse operations to move the 7x to the left-side of the equation as follows:

  • 3x² + 2 = 7x

  • 3x² + 2 (-7x) = 7x (-7x)

  • 3x² + 2 -7x = 0

Notice that result, 3x² + 2 -7x = 0, still isn’t in ax² + bx + c = 0 form. However, the commutative property allows us to rearrange the terms as follows:

  • 3x² + 2 -7x = → 3x² -7x +2

Now we have an equivalent equation, 3x² -7x +2=0, that is in ax² + bx + c = 0 form, where:

  • a=3

  • b=-7

  • c=2

Sometimes you will be given equations that have to be rearranged in order to use the quadratic formula. If you can not rearrange an equation so that it can be expressed in ax² + bx + c = 0 form, then you can not solve it using the quadratic formula.

This example, however, can now be solved using the quadratic formula as follows:

  • x= (-b ± [√(b² - 4ac)]) / 2a

  • x= -(-7) ± [√(-7² - 4(3)(2))]) / 2(3)

  • x= 7 ± [√(49 - 24)] / 6

  • x= 7 ± [√(25)] / 6

  • x= (7 ± 5) / 6

Now we are left with a much easier equation to work with: x= (7 ± 5) / 6. Let’s go ahead and split it into two separate equations to solve it:

  • Plus: x= (7 + 5) / 6

  • Minus: x= (7 - 5) / 6

We can solve for x in each equation as follows:

  • x= (7+5) / 6 = 12/6 = 2 → x=2

  • x= (7-5) / 6 = 2/6 = 1/3 → x=1/3

Final Answer: x=2 and x=1/3

That’s all that there is to it! You can review of the steps to solving this quadratic formula example by looking at the illustration in Figure 07 below.

 

Figure 07: Quadratic Formula Examples: Rearranging an equation to put it into ax² + bx + c = 0 form.

 

Do you need more practice with using the Quadratic Formula?

Check out our free library of Quadratic Formula Worksheets (with answer keys)

Keep Learning:

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