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

This formula is one of the shortcut formulas and it describes the square sum of two numbers.

That is, when we encounter two numbers with a plus sign (sum) and they are between parentheses and raised as an expression to the square, we can use this formula.
Pay attention - The formula also works for non-algebraic expressions or combined combinations with numbers and unknowns.
It's good to know that it is very similar to the formula for the difference of squares and differs only in the minus sign of the central element.

Practice Square of sum

Examples with solutions for Square of sum

Exercise #1

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 #2

(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 #3

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 take a look at 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

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

Now let us consider, what number when multiplied by itself will equal 4 and what number when multiplied by itself will equal 25?

The answers are respectively 2 and 5:

We insert these into the formula:

(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 #4

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

Video Solution

Step-by-Step Solution

We will first solve the exercise by opening the inner brackets:

(2[x+3])²

(2x+6)²

We will then 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 #5

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

Video Solution

Step-by-Step Solution

In order 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 #6

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

Video Solution

Step-by-Step Solution

In order 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 then reinsert them back into the formula as follows:

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 #7

Look at the following square:

AAABBBDDDCCC5+2X

Express the area of the square.

Video Solution

Step-by-Step Solution

Remember that the area of a square is equal to the side of the square squared

The formula for the area of a square is:

S=a2 S=a^2

Now let's substitute the data into the formula:

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

Answer

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

Exercise #8

Given the rectangle ABCD

AB=X

The ratio between AB and BC is x2 \sqrt{\frac{x}{2}}

We mark the length of the diagonal A the rectangle in m

Check the correct argument:

XXXmmmAAABBBCCCDDD

Video Solution

Step-by-Step Solution

Given that:

ABBC=x2 \frac{AB}{BC}=\sqrt{\frac{x}{2}}

Given that AB equals X

We will substitute accordingly in the formula:

xBC=x2 \frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}

x2=BCx x\sqrt{2}=BC\sqrt{x}

x2x=BC \frac{x\sqrt{2}}{\sqrt{x}}=BC

x×x×2x=BC \frac{\sqrt{x}\times\sqrt{x}\times\sqrt{2}}{\sqrt{x}}=BC

x×2=BC \sqrt{x}\times\sqrt{2}=BC

Now let's focus on triangle ABC and use the Pythagorean theorem:

AB2+BC2=AC2 AB^2+BC^2=AC^2

Let's substitute the known values:

x2+(x×2)2=m2 x^2+(\sqrt{x}\times\sqrt{2})^2=m^2

x2+x×2=m2 x^2+x\times2=m^2

We'll add 1 to both sides:

x2+2x+1=m2+1 x^2+2x+1=m^2+1

(x+1)2=m2+1 (x+1)^2=m^2+1

Answer

m2+1=(x+1)2 m^2+1=(x+1)^2

Exercise #9

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

(x+y)2 (x+y)^2

Video Solution

Answer

y2+x2+2xy y^2+x^2+2xy

Exercise #10

Solve for x:

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

Video Solution

Answer

x=0 x=0

Exercise #11

(7+8)2=? (7+8)^2=\text{?}

Video Solution

Answer

72+2×7×8+82 7^2+2\times7\times8+8^2

Exercise #12

(a+b)2=? (a+b)^2=\text{?}

Video Solution

Answer

a2+2ab+b2 a^2+2ab+b^2

Exercise #13

y=x2+9x+24 y=x^2+9x+24

Which expression should be added to y so that:

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

Video Solution

Answer

x+1 x+1

Exercise #14

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

Video Solution

Answer

x4+8x2+16 x^4+8x^2+16

Exercise #15

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

Video Solution

Answer

x2+2x3+x4 x^2+2x^3+x^4

Topics learned in later sections

  1. The formula for the difference of squares
  2. Abbreviated Multiplication Formulas
  3. Multiplication of the sum of two elements by the difference between them