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.


Difference of two squares

When carrying out mathematical operations it is common to find subtractions between two numbers, however, when finding the subtraction of the squares of two numbers we will say that this is a difference of squares and it is of our particular interest because through the distributive property, we can express it as the product of two factors.

Formally, if a and b are two real numbers, then the difference of their squares will be equal to the sum of the first plus the second, multiplied by the subtraction of the first by the second, that is,

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

This type of expression is often found in the development of algebraic operations and is used mainly for factoring operations, let’s see in the following examples how to apply this operation

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Examples

Example 1

Factorize the expression 5^2 - 3^2. Note that in this case, we can simply apply the power of each of the summands and perform the subtraction directly.

5^2 - 3^2 = 25 - 9

= 16

Example 2

Factorize the expression x^2 - 9. We notice that in this case, one of the summands is an x squared and the other one is a nine, so we cannot make the subtraction between them so we apply the difference of squares noting that nine is equal to three squared.

x^2 - 9 = x^2 - 3^2

= (x-3)(x+3)

Example 3

Factorize the expression x^2 - 2. We notice that in this case, one of the summands is an x-squared and the other is two, so we cannot perform the subtraction between them so we apply the difference of squares noting that two can be rewritten as 2 = \left( \sqrt{2} \right)^2.

x^2 - 2 = x^2 -\left( \sqrt{2} \right)^2

= \left(x-\sqrt{2} \right) \left(x+\sqrt{2} \right)

In this way, we can notice that if the square root of a number is not exact, it can be rewritten to use the difference of squares.

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Example 4

Factorize the expression 8 - x^6. We notice that in this case, one of the summands is 8 and the other one is x to six, so we cannot make the subtraction between them so we apply the difference of squares noting that eight can be rewritten as 8 = \left( \sqrt{8} \right)^2 and x to six as x^6 = \left( x^3 \right)^2.

8 - x^6 = \left( \sqrt{8} \right)^2 - \left(x^3 \right)^2

= \left(\sqrt{8}-x^3 \right) \left(\sqrt{8}+x^3 \right)

Example 5

Factorize the expression 36x^4 - 5x^8. We notice that in this case, we cannot make the subtraction between them so we apply the difference of squares using the observations exposed in the previous examples.

36x^4 - 5x^8 = \left( 6x^2 \right)^2 - \left( \sqrt{5}x^4 \right)^2

= \left(6x^2-\sqrt{5}x^4 \right) \left(6x^2+\sqrt{5}x^4 \right)