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.