In this chapter, we will learn to simplify the expression (a + b).(a – b) and present a shortcut formula for fast calculation.

At the end of the chapter, some problems are also given for your practice.

## How to solve expression (x + y) (x – y) ?

To solve the expression, multiply the individual components as given below;

\mathtt{\Longrightarrow \ ( x+y) \ ( x-y)}\\\ \\ \mathtt{\Longrightarrow \ x\ ( x-y) +y\ ( x-y)}\\\ \\ \mathtt{\Longrightarrow \ x^{2} -xy\ +xy\ -\ y^{2}}\\\ \\ \mathtt{\Longrightarrow \ x^{2} -\ y^{2}}

Hence, we get the following formula;

\mathtt{( x+y)( x-y) =\ x^{2} -\ y^{2}}

This is an important formula. Please memorize it as it would help to solve questions faster.

I hope you understood the above concept. Let us solve some problems for our practice.

### Simplifying (x + y) (x – y) – Solved problems

**Example 01**

Simplify expression (5 + a) (5 – a)

**Solution**

Referring the formula;

\mathtt{( x+y)( x-y) =\ x^{2} -\ y^{2}}

Putting the values, we get;

\mathtt{\Longrightarrow \ 5^{2} -\ a^{2}}\\\ \\ \mathtt{\Longrightarrow \ 25\ -\ a^{2}}

Hence, \mathtt{25\ -\ a^{2}} is the solution.

**Example 02**

Simplify the below expression

\mathtt{\left(\frac{a}{2} +\frac{b}{3}\right) \ \left(\frac{a}{2} -\frac{b}{3}\right)}

**Solution**

Referring the formula;

\mathtt{( x+y)( x-y) =\ x^{2} -\ y^{2}}

Here;

x = a / 2

y = b / 3

Putting the values in formula, we get;

\mathtt{\Longrightarrow \ \left(\frac{a}{2}\right)^{2} -\left(\frac{b}{3}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{a^{2}}{4} -\frac{b^{2}}{9}}

Subtracting the fraction using LCM method.

\mathtt{\Longrightarrow \ \frac{9a^{2} -4b^{2}}{36}}

Hence, the above expression is the solution.

**Example 03**

Simplify the below expression.

⟹ (3x + 7y -11) (3x + 7y + 11)

**Solution**

Referring the formula;

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

Where;

a = 3x + 7y

b = 11

Putting the values, we get;

\mathtt{\Longrightarrow \ ( 3x+7y)^{2} -( 11)^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 3x)^{2} +( 7y)^{2} +2( 3x)( 7y) -121}\\\ \\ \mathtt{\Longrightarrow \ 9x^{2} +49y^{2} +42xy-121}

Hence, above expression is the solution.

**Example 04**

Solve the below expression.

\mathtt{\Longrightarrow \ \frac{9x^{2} -49}{( 3x-7)( 5x)}}

**Solution**

Simplifying the expression.

\mathtt{\Longrightarrow \ \frac{( 3x)^{2} -7^{2}}{( 3x-7)( 5x)}}

Referring the formula;

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

Using the formula in numerator, we get;

\mathtt{\Longrightarrow \ \frac{( 3x-7)( 3x+7)}{( 3x-7)( 5x)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\cancel{( 3x-7)}( 3x+7)}{\cancel{( 3x-7)}( 5x)}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3x+7}{5x}}

Hence, the above expression is solution.

**Example 05**

Expand the below expression;

\mathtt{16x^{2} -169y^{2}}

**Solution**

The above expression can be written as;

\mathtt{\Longrightarrow \ ( 4x)^{2} -( 13y)^{2}}

Referring the formula;

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

Here;

a = 4x

b = 13y

Using the above formula;

\mathtt{\Longrightarrow \ ( 4x-13y) \ ( 4x+13y)}

Hence, the above expression is the expanded form.

**Next chapter** : **Cube of sum formula**