Abbreviated Multiplication Formulas

Abbreviated multiplication formulas will be used throughout our math studies, from elementary school to high school. In many cases, we will need to know how to expand or add these equations to arrive at the solution to various math exercises.

Just like other math topics, even in the case of abbreviated multiplication formulas, there is nothing to fear. Understanding the formulas and lots of practice on the topic will give you complete control. So let's get started :)

Abbreviated Multiplication Formulas for 2nd Grade

Here are the basic formulas for abbreviated multiplication:

$(X + Y)^2=X^2+ 2XY + Y^2$

$(X - Y)^2=X^2 - 2XY + Y^2$

$(X + Y)\times (X - Y) = X^2 - Y^2$

Abbreviated Multiplication Formulas for 3rd Grade

$(a+b)^3=a^3+3a^2 b+3ab^2+b^3$

$(a-b)^3=a^3-3a^2 b+3ab^2-b^3$

Abbreviated Multiplication Formulas Verification

We will test the shortcut multiplication formulas by expanding the parentheses.

$(X + Y)^2 = (X + Y)\times (X+Y) =$

$X^2 + XY + YX + Y^2=$

Since: $XY = YX$

$X^2 + 2XY + Y^2$

$(X - Y)^2 = (X - Y)\times (X-Y) =$

$X^2 - XY - YX + Y^2=$

Since: $XY = YX$

$X^2 - 2XY + Y^2$

$(X + Y)\times (X-Y) =$

$X^2 - XY + YX - Y^2=$

Since: $XY = YX$

$- XY + YX = 0$

$X^2 - Y^2$

Abbreviated Multiplication Practice

$(X + 2)^2=X^2 - 8$

$-(X + 2)^2=-X^2 - 8$

$(X + 3)^2=(X-4)\times (X+4)$

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Abbreviated Multiplication Practice Solutions

$(X + 2)^2=X^2 - 8$

$-(X + 2)^2=-X^2 - 8$

$(X + 3)^2=(X-4)\times (X+4)$

Detailed explanation

Start practice

Test yourself on short multiplication formulas!

Choose the expression that has the same value as the following:

$$ (x+y)^2 $$

Practice more now

Abbreviated Multiplication Formulas

Abbreviated multiplication formulas are here to stay, and we will use them in almost all the exercises we encounter in the future.
But hey! Don't stress out. We are lucky to know them because they are the ones that will help us solve exercises correctly and efficiently.
We will separate the three abbreviated multiplication formulas into $4$ categories:

Multiplication of the Sum and Difference of Two Terms
$(X + Y)\times (X - Y) = X^2 - Y^2$
Difference of Squares Formula
$(X - Y)^2=X^2 - 2XY + Y^2$
Sum of Squares Formula
$(X + Y)^2=X^2+ 2XY + Y^2$
Formulas related to two expressions in the $3rd$ power
$(a+b)^3=a^3+3a^2 b+3ab^2+b^3$
$(a-b)^3=a^3-3a^2 b+3ab^2-b^3$

Let's start with the first category:

Multiply the Sum of Two Terms by the Difference Between Them

$(X + Y)\times (X - Y) = X^2 - Y^2$

As you can see, this formula can be used when there is a product between the sum of two specific terms and the difference of those two terms.
Instead of presenting them as a product between a sum and a difference, you can write it $X^2 - Y^2$ .
Similarly, if you are presented with such an expression $X^2 - Y^2$ that represents the difference of two squared numbers, you can write it like this: $(X + Y)\times (X - Y)$
Pay attention: it says in the formula $X$ and $Y$ But it works both in non-algebraic expressions and in expressions that combine variables and numbers.

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Test your knowledge

Question 1

Choose the expression that has the same value as the following:

$$ (x+3)^2 $$

Question 2

Choose the expression that has the same value as the following:

$$ (x-y)^2 $$

Question 3

Choose the expression that has the same value as the following:

$$ (x-7)^2 $$

Let's look at an example.

If we are given:
$(x+2)(x-2)$
We can see that it is a product of the sum of the two terms and the difference of the two terms.
Therefore, we can express the same expression according to the formula in the following way:
$x^2-2^2$
$x^2-4$
Similarly, if we were given the expression:
$x^2-4$
we could express $4$ as a square number, that is $2^2$ ,
to arrive at a form that matches the formula:
$x^2-2^2$
and, therefore, use the formula and express the expression in this way:

$x^2-2^2=(x-2)(x+2)$

Excellent.
Now let's move on to the formula for the difference of squares.

The formula for the difference of squares

$(X - Y)^2=X^2 - 2XY + Y^2$

This formula describes the square of the difference between two numbers, that is, when we encounter two numbers with a minus sign between them, the difference, and they will be in parentheses and appear as an expression squared, we can use this formula.
Keep in mind that, although the formula contains algebraic elements, the formula also works with non-algebraic expressions or combined expressions between numbers and algebra.

Let's look at an example.

$(X-5)^2=$
Here we identify two terms with a minus sign between them, enclosed in parentheses and raised to the square as a single expression.
Therefore, we can use the difference of squares formula.
We will work according to the formula and pay attention to the minus and plus signs.
We will obtain:
$(X-5)^2=x^2-10x+25$
Basically, we have expressed the same expression differently using the formula.
Great. Now let's move on to the sum of squares formula.

Do you know what the answer is?

Question 1

$$ (x^2-6)^2= $$

Question 2

$$ (x-x^2)^2= $$

Question 3

Solve:

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

The formula for the sum of squares

$(X + Y)^2=X^2+ 2XY + Y^2$

This formula describes the square of the sum of two numbers, that is, when we encounter two numbers with a plus sign in the middle, meaning a sum, and they are enclosed in parentheses and appear as a squared expression, we can use this formula.
Keep in mind: although algebraic elements appear in the formula, it also works with non-algebraic expressions or combined expressions between numbers and algebra.
Note: this formula is very similar to the formula for the difference of squares and differs only in the minus sign in the middle term.

Let's look at an example.

$(X+4)^2=$
Here we identify two terms with a plus sign between them, enclosed in parentheses and raised to the square as a single expression.
Therefore, we can use the formula for the sum of squares.
We work according to the formula and pay attention to the minus and plus signs.
We will obtain:
$(X+4)^2=x^2+8x+16$
Basically, we express the same expression differently using the formula.

Now, after you have become deeply familiar with the previous abbreviated multiplication formulas of $2°$ degree, we will move on to the formulas related to two expressions to the 3rd power.

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

$(a+b)^3=a^3+3a^2 b+3ab^2+b^3$ , $(a-b)^3=a^3-3a^2b+3ab^2-b^3$

Here we can also recognize that there are two different formulas for the difference and the sum of terms.
Let's start with the first formula for the sum:
$(a+b)^3=a^3+3a^2 b+3ab^2+b^3$
Here we can also recognize that there are two different formulas for the difference and the sum of terms.
Let's start with the first formula for the sum:

Let's look at an example.

When we are given the following expression:
$(X+2)^3=$
We can identify two terms with a plus sign between them that are enclosed in parentheses and raised to the power of three as a single expression.
Therefore, we can use the corresponding formula.
We will work according to the formula and pay attention to the minus and plus signs.
$(X+2)^3=x^3+3\times x^2\times 2+3\times x\times 2^2+2^3$
$(X+2)^3=x^3+6x^2+12x+8$
Basically, we express the same expression differently with the help of the formula.

Now, let's move on to the second formula for the difference.

$(a-b)^3=a^3-3a^2 b+3ab^2-b^3$

This formula describes a way to express the difference of two terms, when they are enclosed in parentheses and raised as a single expression to the power of three.
The formula can be used with algebraic terms or with numbers and also in combination.

Let's look at an example.

When we are given the following expression:
$(X-4)^3=$
We can identify two terms with a minus sign between them that are enclosed in parentheses and raised to the power of three as a single expression.
Therefore, we can use the corresponding formula.
We will work according to the formula. Pay attention to the minus and plus signs.
$(X-4)^3=x^3-3\times x^2\times 4+3\times x\times 4^2-4^3$
$(X-4)^3=x^3-12x^2+48x-64$
Basically, we express the same expression differently using the formula.

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

Positive, Negative Numbers and Zero

The Real Number Line

Opposite Numbers

Absolute Value

Elimination of Parentheses in Real Numbers

Multiplication and Division of Real Numbers

Multiplicative Inverse

Multiplication Tables

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

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Examples and Exercises with Solutions for Abbreviated Multiplication Formulas

Exercise #1

Choose the expression that has the same value as the following:

$(x+3)^2$

Video Solution

Step-by-Step Solution

We use the abbreviated multiplication formula:

$x^2+2\times x\times3+3^2=$

$x^2+6x+9$

Answer

$x^2+6x+9$

Exercise #2

Choose the expression that has the same value as the following:

$(x-y)^2$

Video Solution

Step-by-Step Solution

We use the abbreviated multiplication formula:

$(x-y)(x-y)=$

$x^2-xy-yx+y^2=$

$x^2-2xy+y^2$

Answer

$x^2-2xy+y^2$

Exercise #3

Solve:

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

Video Solution

Step-by-Step Solution

We use the abbreviated multiplication formula:

$4-x^2=0$

We isolate the terms and extract the root:

$4=x^2$

$x=\sqrt{4}$

$x=\pm2$

Answer

±2

Exercise #4

$x^2=6x-9$

Video Solution

Step-by-Step Solution

Let's solve the given equation:

$x^2=6x-9$

First, let's arrange the equation by moving terms:

$x^2=6x-9 \\ x^2-6x+9=0$

Now let's notice that we can factor the expression on the left side using the perfect square trinomial formula for a binomial squared:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-2\textcolor{red}{a}\textcolor{blue}{b}+\textcolor{blue}{b}^2$

We'll do this using the fact that:

$9=3^2$ Therefore, we'll represent the rightmost term as a squared term:

$x^2-6x+9=0 \\ \downarrow\\ \textcolor{red}{x}^2-6x+\textcolor{blue}{3}^2=0$

Now let's examine again the perfect square trinomial formula mentioned earlier:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

And the expression on the left side in the equation we got in the last step:

$\textcolor{red}{x}^2-\underline{6x}+\textcolor{blue}{3}^2=0$

Notice that the terms $\textcolor{red}{x}^2,\hspace{6pt}\textcolor{blue}{3}^2$ indeed match the form of the first and third terms in the perfect square trinomial formula (which are highlighted in red and blue),

However, in order to factor the expression in question (which is on the left side of the equation) using the perfect square trinomial formula mentioned, the remaining term must also match the formula, meaning the middle term in the expression (underlined with a single line):

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

In other words - we'll ask if we can represent the expression on the left side of the equation as:

$\textcolor{red}{x}^2-\underline{6x}+\textcolor{blue}{3}^2=0\\ \updownarrow\text{?}\\ \textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{3}}+\textcolor{blue}{3}^2=0$

And indeed it is true that:

$2\cdot x\cdot3=6x$

Therefore we can represent the expression on the left side of the equation as a perfect square binomial:

$\textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{3}}+\textcolor{blue}{3}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{3})^2=0$

From here we can take the square root of both sides of the equation (and don't forget that there are two possibilities - positive and negative when taking an even root of both sides of an equation), then we'll easily solve by isolating the variable:

$(x-3)^2=0\hspace{8pt}\text{/}\sqrt{\hspace{6pt}}\\ x-3=\pm0\\ x-3=0\\ \boxed{x=3}$

Let's summarize the solution of the equation:

$x^2=6x-9 \\ x^2-6x+9=0 \\ \downarrow\\ \textcolor{red}{x}^2-2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{3}+\textcolor{blue}{3}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{3})^2=0 \\ \downarrow\\ x-3=0\\ \downarrow\\ \boxed{x=3}$

Therefore the correct answer is answer C.

Answer

$x=3$

Exercise #5

$x^2+144=24x$

Video Solution

Step-by-Step Solution

Let's solve the given equation:

$x^2+144=24x$

First, let's arrange the equation by moving terms:

$x^2+144=24x \\ x^2-24x+144=0$

Now let's notice that we can factor the expression on the left side using the perfect square trinomial formula:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-2\textcolor{red}{a}\textcolor{blue}{b}+\textcolor{blue}{b}^2$

We can do this using the fact that:

$144=12^2$

Therefore, we'll represent the rightmost term as a squared term:

$x^2-24x+144=0 \\ \downarrow\\ \textcolor{red}{x}^2-24x+\textcolor{blue}{12}^2=0$

Now let's examine again the perfect square trinomial formula mentioned earlier:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

And the expression on the left side in the equation we got in the last step:

$\textcolor{red}{x}^2-\underline{24x}+\textcolor{blue}{12}^2=0$

Notice that the terms $\textcolor{red}{x}^2,\hspace{6pt}\textcolor{blue}{12}^2$ indeed match the form of the first and third terms in the perfect square trinomial formula (which are highlighted in red and blue),

However, in order to factor this expression (which is on the left side of the equation) using the perfect square trinomial formula mentioned, the remaining term must also match the formula, meaning the middle term in the expression (underlined):

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

In other words - we'll ask if we can represent the expression on the left side of the equation as:

$\textcolor{red}{x}^2-\underline{24x}+\textcolor{blue}{12}^2=0\\ \updownarrow\text{?}\\ \textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{12}}+\textcolor{blue}{12}^2=0$

And indeed it is true that:

$2\cdot x\cdot12=24x$

Therefore we can represent the expression on the left side of the equation as a perfect square trinomial:

$\textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{12}}+\textcolor{blue}{12}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{12})^2=0$

$(x-12)^2=0\hspace{8pt}\text{/}\sqrt{\hspace{6pt}}\\ x-12=\pm0\\ x-12=0\\ \boxed{x=12}$

Let's summarize the solution of the equation:

$x^2+144=24x \\ x^2-24x+144=0 \\ \downarrow\\ \textcolor{red}{x}^2-2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{12}+\textcolor{blue}{12}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{12})^2=0 \\ \downarrow\\ x-12=0\\ \downarrow\\ \boxed{x=12}$

Therefore the correct answer is answer C.

Answer

$x=12$

Check your understanding

Question 1

$$ (4b-3)(4b-3) $$

Rewrite the above expression as an exponential summation expression:

Question 2

Rewrite the following expression as an addition and as a multiplication:

$$ (3x-y)^2 $$

Question 3

$(a-4)(a-4)=\text{?}$

Start practice

Abbreviated Multiplication Formulas

Abbreviated Multiplication Formulas for 2nd Grade

Abbreviated Multiplication Formulas for 3rd Grade

Abbreviated Multiplication Formulas Verification

Abbreviated Multiplication Practice

Abbreviated Multiplication Practice Solutions

Test yourself on short multiplication formulas!

Abbreviated Multiplication Formulas

Multiply the Sum of Two Terms by the Difference Between Them

Let's look at an example.

The formula for the difference of squares

Let's look at an example.

The formula for the sum of squares

Let's look at an example.

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

Let's look at an example.

Let's look at an example.

Examples and Exercises with Solutions for Abbreviated Multiplication Formulas

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Square of sum

Square of Difference

Difference of squares

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