Math Conjugates: Definition & Explanation
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
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