# The Conjugate of a Sum

Next we will define an expression that is closely related to the difference of squares, because when we find the addition (or subtraction as the case may be) of two real numbers, we can define an expression that will allow us to write that subtraction as a difference of squares.

Formally, if $a$ and $b$ are two real numbers, the conjugate of the sum $(a+b)$ is defined as $(a-b)$. Similarly, the conjugate of the subtraction $(a-b)$ is defined as $(a+b)$. That is, the sign between the two is changed. The importance of the conjugate lies in the fact that the product of an addition by its conjugate is equal to a difference of squares, that is,

This equality can be deduced by performing the distributive property of the real numbers, let’s see then,

This type of expressions is often found in the development of algebraic operations and is used mainly to simplify operations, let’s see in the following examples how to identify the conjugation of some expressions:

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### Examples

#### Example 1

Identify the conjugate of $12 - 5$. It does not have much sense to identify the conjugate of this expression because we can simply make the subtraction and obtain 7 as a result.

#### Example 2

Identify the conjugate of $\sqrt{12} - 5$. Note that one of the involved sums is the square root of twelve, so it cannot be subtracted with five, so we conclude that its conjugate is $\sqrt{12} + 5$.

#### Example 3

Identify the conjugate of $3 + \sqrt{8}$. Note that one of the summands involved is the square root of eight, so it cannot be added with three, so we conclude that its conjugate is $3 - \sqrt{8}$.

#### Example 4

Identify the conjugate of $3x - 7$. Note that one of the sums involved is three multiplied by one unknown, so it cannot be subtracted with seven, then, we conclude that its conjugate is $3x + 7$.

#### Example 5

Identify the conjugate of $15 + 4x$. Let’s notice that one of the involved sums is four multiplied by one unknown, therefore it cannot be added with 15, then, we conclude that its conjugate is $15 - 4x$.

#### Example 6

Identify the conjugate of $6 + \sqrt{x+2}$. This subtraction cannot be done, so we conclude that your conjugate is $6 - \sqrt{x+2}$. Noting that the sign inside the root does not change.

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