Multiplication of the sum of two elements by the difference between them

🏆Practice multiplication of the sum of two terms by the difference between them

(X+Y)×(XY)=X2Y2(X + Y)\times (X - Y) = X^2 - Y^2

This is one of the shortened multiplication formulas.

As can be seen, this formula can be used when there is a multiplication between the sum of two particular elements and the subtraction between the two elements.
Instead of presenting them as a multiplication of sum and subtraction, it can be written X2Y2X^2 - Y^2 and it expresses exactly the same thing. In the same way, if such an expression X2Y2X^2 - Y^2 representing the subtraction of two squared numbers is presented to you, you can write it like this: (X+Y)×(XY)(X + Y)\times (X - Y)
Pay attention: the formula works both in non-algebraic expressions and in expressions that combine unknowns and numbers.

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Test yourself on multiplication of the sum of two terms by the difference between them!

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Solve:

\( (2+x)(2-x)=0 \)

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Let's look at an example

If we are given: (x+4)(x4)(x+4)(x-4)
We can see that we are referring to a multiplication between the sum of two elements and the difference between them.
Therefore, we can present the same expression according to the formula in the following way:
x242x^2-4^2
x216x^2-16
In the same way, if we were given the expression:
x216x^2-16
We could express 1616 as a squared number, that is 424^2 ,
Obtain a representation that fits the formula:
x242x^2-4^2
From here using the formula and presenting the expression in the following way:

x242=(X4)(x+4)x^2-4^2=(X-4)(x+4)


If you are interested in this article, you might also be interested in the following articles:

The formula for the difference of squares

The formula for the sum of squares

The formulas that refer to two expressions to the power of 3

In the blog of Tutorela you will find a variety of articles about mathematics.


Examples and exercises with solutions for multiplying the sum of two elements by the difference between them

Exercise #1

Solve:

(2+x)(2x)=0 (2+x)(2-x)=0

Video Solution

Step-by-Step Solution

We use the abbreviated multiplication formula:

4x2=0 4-x^2=0

We isolate the terms and extract the root:

4=x2 4=x^2

x=4 x=\sqrt{4}

x=±2 x=\pm2

Answer

±2

Exercise #2

(2x)23=6 (2x)^2-3=6

Video Solution

Step-by-Step Solution

We move the sections and equal to 0

4x236=0 4x^2-3-6=0

4x29=0 4x^2-9=0

We use the shortcut multiplication formula:

4(x294)=0 4(x^2-\frac{9}{4})=0

x2(32)2=0 x^2-(\frac{3}{2})^2=0

(x32)(x+32)=0 (x-\frac{3}{2})(x+\frac{3}{2})=0

x=±32 x=\pm\frac{3}{2}

Answer

±32 ±\frac{3}{2}

Exercise #3

Complete the following exercise:

(x+12)(x12)=0 (\sqrt{x}+\frac{1}{2})(\sqrt{x}-\frac{1}{2})=0

Video Solution

Answer

14 \frac{1}{4}

Exercise #4

Solve the exercise:

(x+3)(x3)+(x+1)(x1)=0 (x+3)(x-3)+(x+1)(x-1)=0

Video Solution

Answer

±5 ±\sqrt{5}

Exercise #5

Fill in the missing element to obtain a true expression:

(x+)(x)=x2121 (x+_—)\cdot(x-_—)=x^2-121

Video Solution

Answer

11

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