# The sum of squares formula - Examples, Exercises and Solutions

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

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:

$(x+3)^2$

### Step-by-Step Solution

We use the abbreviated multiplication formula:

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

$x^2+6x+9$

$x^2+6x+9$

### Exercise #2

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

### Step-by-Step Solution

According to the shortened multiplication formula:

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

$(7+x)^2$

$(7+x)^2$

### Exercise #3

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

### Step-by-Step Solution

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

Let's remember the formulas:

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

$(x+y)^2=x^2+2xy+y^2$

$(x+y)\times(x-y)=x^2-y^2$

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:

$(2x+5)^2=$

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

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

$4x^2+20x+25$

That means our solution is correct.

$(2x+5)^2$

### Exercise #4

$(2\lbrack x+3\rbrack)^2=$

### 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

$4x^2+24x+36$

### Exercise #5

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

### Step-by-Step Solution

To solve the exercise, remember the abbreviated multiplication formulas:

$(x+y)^2=x^2+2xy+y^2$

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

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

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

We will place them back in the formula:

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

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

$5x^2+24x+30$

$5x^2+24x+30$

### Exercise #1

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

### 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:

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

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

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

x²+2x+1+x²+4x+4=
2x²+6x+5

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

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

### Exercise #2

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

$(x+y)^2$

### Video Solution

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

### Exercise #3

Solve for x:

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

### Video Solution

$x=0$

### Exercise #4

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

### Video Solution

$a^2+2ab+b^2$

### Exercise #5

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

### Video Solution

$7^2+2\times7\times8+8^2$

### Exercise #1

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

Which expression should be added to y so that:

$y=(x+5)^2$

### Video Solution

$x+1$

### Exercise #2

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

### Video Solution

$x^4+8x^2+16$

### Exercise #3

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

### Video Solution

$x^2+2x^3+x^4$

### Exercise #4

What is the value of x?

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

### Video Solution

$x=1$

### Exercise #5

$(x+2)^2-12=x^2$

### Video Solution

$x=2$