Abbreviated Multiplication Formulas

What are Abbreviated Multiplication Formulas?

Abbreviated multiplication formulas, also known as algebraic identities, are shortcuts that simplify the process of expanding and factoring expressions. These formulas save time and reduce the steps required for complex calculations. 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.

Abbreviated Multiplication Formulas for 2nd degree

Here are the basic formulas for abbreviated multiplication:

The square of the sum:
$(X + Y)^2=X^2+ 2XY + Y^2$

The squared difference:

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

The difference of squares:

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

Abbreviated Multiplication Formulas for 3rd degree

Abbreviated multiplication formulas for 3rd-degree expressions, also known as cubic identities, build upon the concepts of the 2nd-degree formulas we’ve already covered. The key difference is the adjustment for working with cubic (3rd-degree) terms instead of quadratic (2nd-degree) terms. These formulas simplify complex cubic expressions, breaking them into manageable parts to make calculations faster and more efficient. They are particularly useful in solving problems involving volumes of cubes and other 3D shapes or in advanced mathematics, such as polynomial factoring and equation solving.

Here are two of the most common abbreviated multiplication formulas for 3rd-degree expressions:

Cube of a Sum

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

Cube of a Difference

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

Abbreviated Multiplication Formulas Verification and Examples

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

Square of a Sum:

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

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

Since: $XY = YX$

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

Example:

$(x+3)^2$

$(x+3)^2=x^2+2(x)(3)+3^2=x^2+6x+9$

Square of a Difference:

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

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

Since: $XY = YX$

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

Example:

$(x−4)^2$

$(x−4)^2=x^2−2(x)(4)+4^2=x^2−8x+16$

Product of a Sum and a Difference:

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

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

Since: $XY = YX$

$- XY + YX = 0$

$X^2 - Y^2$

Example:

$(x+5)(x−5)$

$(x+5)(x−5)=x^2−5^2=x^2−25$

Using Abbreviated Multiplication Formulas to Shift the Expression Both Ways

It’s important to remember that these formulas are not one-sided; you can use them to switch between different forms of an expression as needed. For example, you can use the formula:

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

to expand an expression in parentheses into its expanded form. Conversely, if you encounter an expression like

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

you can factor it back into

$(X + Y)^2$

This flexibility allows you to work with the representation that is most useful for the problem at hand, whether it’s simplifying, solving, or analyzing the expression. Understanding this two-way functionality is essential for mastering algebraic manipulation.

Detailed explanation

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Test yourself on short multiplication formulas!

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

Rewrite the above expression as an exponential summation expression:

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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

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

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

Question 2

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

Question 3

Declares the given expression as a sum

$$ (7b-3x)^2 $$

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

Solve:

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

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:

$$ (x+3)^2 $$

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.

Abbreviated Multiplication Practice

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

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

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

Check your understanding

Question 1

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

$$ (x-y)^2 $$

Question 2

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

$$ (x-7)^2 $$

Question 3

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

Abbreviated Multiplication Practice Solutions

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

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

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

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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

Solve the following problem:

$x^2=6x-9$

Video Solution

Step-by-Step Solution

Proceed to solve the given equation:

$x^2=6x-9$

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

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

Note that we can factor the expression on the left side by using the perfect square trinomial formula for a binomial squared:

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

As shown below:

$9=3^2$ 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 that we obtained in the last step:

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

Note 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 will query whether 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

Solve the following problem:

$x^2+144=24x$

Video Solution

Step-by-Step Solution

Proceed to solve the given equation:

$x^2+144=24x$

Arrange the equation by moving terms:

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

Note that we are able to factor the expression on the left side by using the perfect square trinomial formula:

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

As demonstrated below:

$144=12^2$

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 once 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 that we obtained in the last step:

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

Note 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 query whether 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

$x=12$

Do you think you will be able to solve it?

Question 1

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

Question 2

Solve for x:

$$ (x+3)^2=x^2+9 $$

Question 3

Solve the following problem:

$$ x^2+144=24x $$

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Abbreviated Multiplication Formulas

What are Abbreviated Multiplication Formulas?

Abbreviated Multiplication Formulas for 2nd degree

Abbreviated Multiplication Formulas for 3rd degree

Abbreviated Multiplication Formulas Verification and Examples

Using Abbreviated Multiplication Formulas to Shift the Expression Both Ways

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.

Abbreviated Multiplication Practice

Abbreviated Multiplication Practice Solutions

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 $3$