Math Conjugates: Definition & Explanation

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Opposite Reciprocals: Definition & Concept

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:02 Conjugate Concept
  • 0:20 What is a Math Conjugate?
  • 0:52 Difference of Squares
  • 1:33 Conjugates with Radicals
  • 3:41 Math Conjugates Sightings
  • 4:26 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Timeline
Autoplay
Autoplay
Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: David Liano
After completing this lesson, you will be able to describe the concept of math conjugates. You will also be able to write math conjugates and use them appropriately to solve problems.

Conjugate Concept

The term conjugate means a pair of things joined together. These two things are exactly the same except for one pair of features that are actually opposite of each other. If you look at these faces, you will notice that they are the same except that they have opposite facial expressions: one has a smile and the other has a frown.

smiley face

smiley face

What is a Math Conjugate?

A math conjugate is formed by changing the sign between two terms in a binomial. For instance, the conjugate of x + y is x - y. We can also say that x + y is a conjugate of x - y. In other words, the two binomials are conjugates of each other. Instead of smile and a frown, math conjugates have a positive sign and a negative sign, respectively.

Let's consider a simple example. The conjugate of 5x + 9 is 5x - 9.

Difference of Squares

Let's now take the conjugates of x + 4 and x - 4 and multiply them together as follows:

(x + 4)(x - 4) = x^2 - 4x + 4x - 16 = x^2 - 16

Notice that two terms, -4x and 4x, cancel each other out during the simplifying process. We are left with a difference of two squares. In fact, the factored form of a difference of two squares is always a pair of conjugates. This concept is usually shown in algebra textbooks as the equation in Figure 1.

Figure 1
difference of squares: (a+b)(a-b) = a^2 - b^2

Conjugates with Radicals

Perhaps a conjugate's most useful function is as a tool when simplifying expressions with radicals, or square roots. Let's first multiply the conjugates shown in Figure 2

Figure 2
conjugate 1

By multiplying the conjugates in Figure 2, we are able to eliminate the radical expressions. In fact, our solution is a rational expression, in this case a natural number. It is usually easier to work with rational numbers instead of irrational numbers.

We cannot just go around and change the value of expressions so that we can get rid of radicals. There needs to be some logical or practical reason. For instance, multiplying an expression by its conjugate is very useful when simplifying certain fractions.

Let's consider the fraction in Figure 3. This fraction is not simplified because there is a radical in the denominator. A radical in the numerator is all right, but not in the denominator. We need to get rid of the square root of 7 from the denominator. One reason for this rule is that fractions are usually easier to add and subtract when the denominator is a rational number.

Figure 3
conjugate in denominator

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support