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=X2+2XY+Y2(X + Y)^2=X^2+ 2XY + Y^2

(XY)2=X22XY+Y2(X - Y)^2=X^2 - 2XY + Y^2

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


Abbreviated Multiplication Formulas for 3rd Grade

(a+b)3=a3+3a2b+3ab2+b3(a+b)^3=a^3+3a^2 b+3ab^2+b^3

​​​​​​​(ab)3=a33a2b+3ab2b3​​​​​​​(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)×(X+Y)=(X + Y)^2 = (X + Y)\times (X+Y) =

X2+XY+YX+Y2=X^2 + XY + YX + Y^2=

Since: XY=YXXY = YX

X2+2XY+Y2X^2 + 2XY + Y^2


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

X2XYYX+Y2=X^2 - XY - YX + Y^2=

Since:XY=YX XY = YX

X22XY+Y2X^2 - 2XY + Y^2


(X+Y)×(XY)=(X + Y)\times (X-Y) =

X2XY+YXY2=X^2 - XY + YX - Y^2=

Since: XY=YX XY = YX

XY+YX=0 - XY + YX = 0

X2Y2X^2 - Y^2


Abbreviated Multiplication Practice

```html

(X+2)2=X28(X + 2)^2=X^2 - 8

(X+2)2=X28-(X + 2)^2=-X^2 - 8

(X+3)2=(X4)×(X+4)(X + 3)^2=(X-4)\times (X+4)


```

Abbreviated Multiplication Practice Solutions

```html

(X+2)2=X28(X + 2)^2=X^2 - 8

(X+2)2=X28-(X + 2)^2=-X^2 - 8

(X+3)2=(X4)×(X+4)(X + 3)^2=(X-4)\times (X+4)


```

Practice Abbreviated multiplication formulas

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

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


(x+3)2 (x+3)^2

Video Solution

Step-by-Step Solution

We use the abbreviated multiplication formula:

x2+2×x×3+32= x^2+2\times x\times3+3^2=

x2+6x+9 x^2+6x+9

Answer

x2+6x+9 x^2+6x+9

Exercise #3

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

(xy)2 (x-y)^2

Video Solution

Step-by-Step Solution

We use the abbreviated multiplication formula:

(xy)(xy)= (x-y)(x-y)=

x2xyyx+y2= x^2-xy-yx+y^2=

x22xy+y2 x^2-2xy+y^2

Answer

x22xy+y2 x^2-2xy+y^2

Exercise #4

(7+x)(7+x)=? (7+x)(7+x)=\text{?}

Video Solution

Step-by-Step Solution

According to the shortened multiplication formula:

Since 7 and X appear twice, we raise both terms to the power:

(7+x)2 (7+x)^2

Answer

(7+x)2 (7+x)^2

Exercise #5

4x2+20x+25= 4x^2+20x+25=

Video Solution

Step-by-Step Solution

In this task, we are asked to simplify the formula using the abbreviated multiplication formulas.

Let's remember the formulas:

(xy)2=x22xy+y2 (x-y)^2=x^2-2xy+y^2

 (x+y)2=x2+2xy+y2 (x+y)^2=x^2+2xy+y^2

(x+y)×(xy)=x2y2 (x+y)\times(x-y)=x^2-y^2

Given that in the given exercise there is only addition operation, the appropriate formula is the second one:

Now let's try to think, what number multiplied by itself will equal 4 and what number multiplied by itself will equal 25?

The answers are respectively 2 and 5:

We will write:

(2x+5)2= (2x+5)^2=

(2x+5)(2x+5)= (2x+5)(2x+5)=

2x×2x+2x×5+2x×5+5×5= 2x\times2x+2x\times5+2x\times5+5\times5=

4x2+20x+25 4x^2+20x+25

That means our solution is correct.

Answer

(2x+5)2 (2x+5)^2

Exercise #1

(2[x+3])2= (2\lbrack x+3\rbrack)^2=

Video Solution

Step-by-Step Solution

First, we will solve the exercise by opening the inner brackets:

(2[x+3])²

(2x+6)²

Now we will use the shortcut multiplication formula:

(X+Y)²=+2XY+

(2x+6)² = 2x² + 2x*6*2 + 6² = 2x+24x+36

Answer

4x2+24x+36 4x^2+24x+36

Exercise #2

(x2)2+(x3)2= (x-2)^2+(x-3)^2=

Video Solution

Step-by-Step Solution

To solve the question, we need to know one of the shortcut multiplication formulas:

(xy)2=x22xy+y2 (x−y)^2=x^2−2xy+y^2

Now, we apply this property twice:

(x2)2=x24x+4 (x-2)^2=x^2-4x+4

(x3)2=x26x+9 (x-3)^2=x^2-6x+9

Now we add:

x24x+4+x26x+9= x^2-4x+4+x^2-6x+9=

2x210x+13 2 x^2-10x+13

Answer

2x210x+13 2x^2-10x+13

Exercise #3

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

Video Solution

Step-by-Step Solution

To solve the exercise, remember the abbreviated multiplication formulas:

(x+y)2=x2+2xy+y2 (x+y)^2=x^2+2xy+y^2

Let's start by using the property in both cases:

(x+3)2=x2+6x+9 (x+3)^2=x^2+6x+9

(x+2)2=x2+4x+4 (x+2)^2=x^2+4x+4

We will place them back in the formula:

2(x2+6x+9)+3(x2+4x+4)= 2(x^2+6x+9)+3(x^2+4x+4)=

2x2+12x+18+3x2+12x+12= 2x^2+12x+18+3x^2+12x+12=

5x2+24x+30 5x^2+24x+30

Answer

5x2+24x+30 5x^2+24x+30

Exercise #4

(x+1)2+(x+2)2= (x+1)^2+(x+2)^2=

Video Solution

Step-by-Step Solution

To solve the exercise, we will need to know the abbreviated multiplication formula:

In this exercise, we will use the formula twice:

(x+1)2=x2+2x+1 (x+1)^2=x^2+2x+1

(x+2)2=x2+4x+4 (x+2)^2=x^2+4x+4

Now, we add:

x2+2x+1+x2+4x+4=2x2+6x+5 x^2+2x+1+x^2+4x+4=2x^2+6x+5

x²+2x+1+x²+4x+4=
2x²+6x+5

Note that a common factor can be extracted from part of the digits: 2(x2+3x)+5 2(x^2+3x)+5

Answer

2(x2+3x)+5 2(x^2+3x)+5

Exercise #5

(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 #1

Look at the square below:

AAABBBDDDCCCX-7

Express its area in terms of x x .

Video Solution

Step-by-Step Solution

Remember that the area of the square is equal to the side of the square raised to the 2nd power.

The formula for the area of the square is

A=L2 A=L^2

We place the data in the formula:

A=(x7)2 A=(x-7)^2

Answer

(x7)2 (x-7)^2

Exercise #2

Declares the given expression as a sum

(7b3x)2 (7b-3x)^2

Video Solution

Answer

49b242bx+9x2 49b^2-42bx+9x^2

Exercise #3

(4b3)(4b3) (4b-3)(4b-3)

Write the above as a power expression and as a summation expression.

Video Solution

Answer

(4b3)2 (4b-3)^2

16b224b+9 16b^2-24b+9

Exercise #4

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

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

Video Solution

Answer

9x26xy+y2 9x^2-6xy+y^2

(3xy)(3xy) (3x-y)(3x-y)

Exercise #5

(a4)(a4)=? (a-4)(a-4)=\text{?}

Video Solution

Answer

a28a+16 a^2-8a+16