The sum of squares formula - Examples, Exercises and Solutions

(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 The sum of squares formula

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 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 #4

(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 #5

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 #1

(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 #2

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 #3

Solve for x:

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

Video Solution

Answer

x=0 x=0

Exercise #4

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

Video Solution

Answer

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

Exercise #5

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

Video Solution

Answer

72+2×7×8+82 7^2+2\times7\times8+8^2

Exercise #1

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 #2

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

Video Solution

Answer

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

Exercise #3

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

Video Solution

Answer

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

Exercise #4

What is the value of x?

(x+3)2=x2+15 (x+3)^2=x^2+15

Video Solution

Answer

x=1 x=1

Exercise #5

(x+2)212=x2 (x+2)^2-12=x^2

Video Solution

Answer

x=2 x=2

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