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