# Difference of Squares

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

## Difference of Square formula

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

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

Hence difference of square of a & b is equal to product of subtraction & addition of the given terms.

The formula is very important as it would help to solve different algebra problems. I strongly urge you to memorize the formula for your examination.

### Proving Difference of Square equation

Let a & b are the two given entities.

The formula is given as;

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

Let’s simplify the right side of equation and check if gets equal to the left side.

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

Hence, LHS = RHS.
We have validated the equation.

Proof 2

Consider the expression \mathtt{9^{2} -5^{2}}

Finding the value using simple calculation

\mathtt{\Longrightarrow \ 9^{2} -5^{2}}\\\ \\ \mathtt{\Longrightarrow \ 81-\ 25}\\\ \\ \mathtt{\Longrightarrow \ 56}

Hence, 56 is the value of given expression.

Now find the value using difference of squares formula.

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

Putting the values;

\mathtt{\Longrightarrow \ 9^{2} -5^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 9-5)( 9+5) \ }\\\ \\ \mathtt{\Longrightarrow \ 4\ .\ 14}\\\ \\ \mathtt{\Longrightarrow \ 56}

Using the formula we get the same value 56.

Hence, the above formula is valid.

## Difference of Square – Solved Problems

Example 01
Expand \mathtt{x^{2} -49}

Solution
The expression can be written as;

\mathtt{\Longrightarrow \ x^{2} -49}\\\ \\ \mathtt{\Longrightarrow \ x^{2} -7^{2}}

We will use formula;
\mathtt{a^{2} -b^{2} =\ ( a-b) \ ( a+b)}

Putting the values;
\mathtt{\Longrightarrow \ x^{2} -7^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( x-7)( x+7) \ }

Hence, \mathtt{( x-7)( x+7) \ } is the expanded form of given expression.

Example 02
Expand \mathtt{y^{4} -144}

Solution
The expression can be written as;

\mathtt{\Longrightarrow \ y^{4} -144}\\\ \\ \mathtt{\Longrightarrow \ \left( y^{2}\right)^{2} \ -\ ( 12)^{2}}

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

Putting the values;

\mathtt{\Longrightarrow \ \left( y^{2}\right)^{2} \ -\ ( 12)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \left( y^{2} -7\right)\left( y^{2} +7\right) \ }

Hence, \mathtt{\ \left( y^{2} -7\right)\left( y^{2} +7\right)} is the expanded form of given expression.

Example 03
Compress the expression \mathtt{( 5x+3y)( 5x-3y) \ }

Solution
\mathtt{( 5x+3y)( 5x-3y) \ } is the expression.

The expression resembles the form of (a + b).( a – b).

Where;
a = 5x
b = 3y

We will use formula
\mathtt{( a+b) \ ( a-b) \ =\ a^{2} -b^{2}}

Putting the values;

\mathtt{\Longrightarrow \ ( 5x+3y)( 5x-3y) \ }\\\ \\ \mathtt{\Longrightarrow \ ( 5x)^{2} -( 3y)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 25x^{2} -9y^{2}}

Hence, \mathtt{25x^{2} -9y^{2}} is the compressed form of given expression.

Example 04
Compress the expression \mathtt{\left( x^{2} +14\right)\left( x^{2} -14\right) \ }

Solution
\mathtt{\left( x^{2} +14\right)\left( x^{2} -14\right) \ } is the expression.

The expression is in form of (a + b). (a – b)

Where;
a = \mathtt{x^{2}}
b = 14

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

Putting the values;
\mathtt{\Longrightarrow \ \left( x^{2} +14\right)\left( x^{2} -14\right) \ }\\\ \\ \mathtt{\Longrightarrow \ \left( x^{2}\right)^{2} -( 14)^{2}}\\\ \\ \mathtt{\Longrightarrow \ x^{4} -196}

Hence, \mathtt{x^{4} -196} is the compressed form of given expression.

Example 05
Find \mathtt{( 10)^{2} \ -\ ( 8)^{2}} using difference of square formula.

Solution
We will use formula;
\mathtt{a^{2} -b^{2} =\ ( a-b) \ ( a+b)}

Putting the values;
\mathtt{\Longrightarrow \ ( 10)^{2} \ -\ ( 8)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 10+8)( 10-8) \ }\\\ \\ \mathtt{\Longrightarrow \ ( 18) \ .( 2)}\\\ \\ \mathtt{\Longrightarrow \ 36}

Hence, 36 is the value of given expression.

Example 06
Express 17 x 23 as difference of squares and find the value.

Solution
17 x 23 can be expressed as;

⟹ 17 x 23

⟹ (20 – 3) (20 + 3)

The expression is in form of (a + b) (a – b)

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

Putting the values;

\mathtt{\Longrightarrow \ ( 20-3)( 20+3) \ }\\\ \\ \mathtt{\Longrightarrow \ ( 20)^{2} \ -( 3){^{2}}}\\\ \\ \mathtt{\Longrightarrow \ 400\ -\ 9}\\\ \\ \mathtt{\Longrightarrow \ 391}

Hence, 391 is the value of expression.

Example 07
Expand \mathtt{2x^{2} -\ 50}

Solution
Here 2 is the common factor.
Simplifying the given expression, we get;

\mathtt{\Longrightarrow \ 2x^{2} -\ 50}\\\ \\ \mathtt{\Longrightarrow \ 2\ \left( x^{2} -\ 25\right)}

Here the term \mathtt{\left( x^{2} -\ 25\right)} is in the form of \mathtt{a^{2} -b^{2}}

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

Solving the given expression;
\mathtt{\Longrightarrow \ 2\ \left( x^{2} -\ 25\right)}\\\ \\ \mathtt{\Longrightarrow \ 2\ ( x-5) \ ( x+5)}

Hence, \mathtt{2\ ( x-5) \ ( x+5)} is the extended form of expression.

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