After + 4x – 16 = x^2

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.

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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

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 xy. We can also say that x + y is a conjugate of xy. 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 – 16Notice 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

We now know that we could multiply the denominator by its conjugate to get rid of the square root of 7; however, that would change the value of the fraction. But we should also know that when we multiply the numerator and denominator of a fraction by the same number or expression, we end up with a fraction that is equivalent to the original fraction. For example, let’s start with the fraction of 1/2. Next, multiply the numerator and denominator by 3:(1 x 3)/(2 x 3) = 3/6.We end up with a fraction that is equivalent to 1/2. Therefore, in our problem in Figure 3, we must also multiply the numerator by the conjugate of the denominator as shown in Figure 4.

Figure 4
simplify with conjugates

If we complete the multiplication and subsequent simplifying, our solution will be as shown in Figure 5. There is no longer a radical in the denominator, and this is what we want.

Figure 5
Quadratic Formula
quadratic formula

The numerator of the quadratic formula is a pair of conjugates. Sometimes these are complex conjugates, but that is getting outside the focus of this lesson.

We might also see conjugates in trigonometry; for instance, the conjugate of 1 – cosx is 1 + cosx. Using conjugates in trigonometry are beneficial when you need to evaluate and simplify trigonometric identities. Again, this is getting outside the focus of this lesson, but knowing the concept of math conjugates might be of assistance when learning new topics.

Lesson Summary

Let’s review:Math conjugates are a simple concept, but are valuable when simplifying some types of fractions. A pair of conjugates is a pair of binomials that are exactly the same except that the signs between the terms are opposite. To create a conjugate of a binomial, just rewrite it and change the sign of the second term.

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