# Square of Difference

In this chapter we will learn the square of difference formula and will solve some questions related to the formula.

## Square of Difference Formula

If a & b are two entities then square of a – b is given by following formula;

\mathtt{( a\ -\ b)^{2} =\ a^{2} +\ b^{2} -2ab}

Hence, the square of a – b is equal to sum of square of individual terms and subtraction of 2ab.

Make sure to memorize the formula as it would help to solve variety of algebra problems.

### Derive Square of Difference Formula

Let a & b are two given terms.

The square of difference of a & b can be expressed as;

\mathtt{( a-b)^{2} =( a-b) .( a-b)}

Simplifying the equation;

\mathtt{\Longrightarrow ( a-b) .( a-b)}\\\ \\ \mathtt{\Longrightarrow \ a^{2} -ab-ab+b^{2}}\\\ \\ \mathtt{\Longrightarrow \ a^{2} -2ab+b^{2}}

Hence, the formula derived is;

\mathtt{( a-b)^{2} \ =\ a^{2} -2ab+b^{2}}

Note:
The formula is also called square of difference of binomial as the expression (a – b) is a binomial with two terms.

### Proof of Square of Difference

Consider the number \mathtt{8-3)^{2}}

Find the value of number using simple calculation;

\mathtt{\Longrightarrow \ ( 8-3)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 5)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 25}

Hence, 25 is the value of given number.

Now let’s find value using Square of Difference formula.

Using the formula;
\mathtt{( a-b)^{2} \ =\ a^{2} -2ab+b^{2}}

Putting the values;

\mathtt{\Longrightarrow \ ( 8-3)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 8^{2} -2( 8)( 3) \ +3^{2}}\\\ \\ \mathtt{\Longrightarrow \ 64-48+9}\\\ \\ \mathtt{\Longrightarrow \ 25}

Using the formula we get the same value 25.

Hence, the above formula is valid.

## Square of Difference – Solved Examples

Example 01
Expand \mathtt{( 9y-2x)^{2}}

Solution
Using the formula;
\mathtt{( a-b)^{2} \ =\ a^{2} -2ab+b^{2}}

Putting the values;

\mathtt{\Longrightarrow \ ( 9y-2x)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 9y){^{2}} -2( 9y)( 2x) \ +( 2x)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 81y^{2} -36xy+4x^{2}}

Hence, \mathtt{\Longrightarrow \ 81y^{2} -36xy+4x^{2}} is the expanded form of expression.

Example 02
Expand \mathtt{\left( y^{3} -10\right)^{2}}

Solution
Using the formula;
\mathtt{( a-b)^{2} \ =\ a^{2} -2ab+b^{2}}

Putting the values we get;

\mathtt{\Longrightarrow \ \left( y^{3} -10\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \left( y^{3}\right){^{2}} -2\left( y^{3}\right)( 10) \ +( 10)^{2}}\\\ \\ \mathtt{\Longrightarrow \ y^{6} -20y^{3} +100}

Hence, \mathtt{y^{6} -20y^{3} +100} is the required solution.

Example 03
Find the value of \mathtt{( 10.7)^{2}} using square of difference formula.

Solution
\mathtt{( 10.7)^{2}} can be written as;

\mathtt{( 10.7)^{2} \Longrightarrow \ ( 11-0.3)^{2}}

Now using the formula;
\mathtt{( a-b)^{2} \ =\ a^{2} -2ab+b^{2}}

Putting the values;

\mathtt{\Longrightarrow \ ( 11){^{2}} -2( 11)( 0.3) \ +( 0.3)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 121-6.6+\ 0.09}\\\ \\ \mathtt{\Longrightarrow \ 114.49}

Hence, 114.49 is the value of given number.

Example 04
Find the value of \mathtt{(18)^{2}} using square of difference formula.

Solution
\mathtt{(18)^{2}} can be written as;

\mathtt{( 18)^{2} \Longrightarrow \ ( 20-2)^{2}}

Now we will use the formula;
\mathtt{( a-b)^{2} \ =\ a^{2} -2ab+b^{2}}

Putting the values;
\mathtt{\Longrightarrow \ ( 20-2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 20)^{2} -2( 20)( 2) \ +( 2)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 400-80+\ 4}\\\ \\ \mathtt{\Longrightarrow \ 324}

Hence, 324 is the value of given expression.

Example 05
Find the value of \mathtt{(999)^{2}} using square of difference formula.

Solution
\mathtt{(999)^{2}} can be written as;

\mathtt{( 999)^{2} \ \Longrightarrow \ ( 1000-1)^{2}}

Now we will use the formula;
\mathtt{( a-b)^{2} \ =\ a^{2} -2ab+b^{2}}

Putting the values;
\mathtt{\Longrightarrow \ ( 1000-1)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 1000)^{2} -2( 1000)( 1) \ +( 1)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 1000000-2000+\ 1}\\\ \\ \mathtt{\Longrightarrow \ 998001}

Hence, 9801 is the solution.

Example 06
Expand \mathtt{\left( 3x^{3} -2y^{2}\right)^{2}}

Solution
The expression is in the form of square of difference.

So we will use the formula;
\mathtt{( a-b)^{2} \ =\ a^{2} -2ab+b^{2}}

Putting the values, we get;

\mathtt{\Longrightarrow \ \left( 3x^{3} -2y^{2}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \left( 3x^{3}\right)^{2} -2\left( 3x^{3}\right)\left( 2y^{2}\right) \ +\left( 2y^{2}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow 9x^{6} -12x^{3} y^{2} +\ 4y^{4}}

Hence, \mathtt{9x^{6} -12x^{3} y^{2} +\ 4y^{4}} is the required expression.