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.