Question

Simplify the following algebraic identities.
${{\left( a+b \right)}^{2}}-{\left( a-b \right)^{2}}=\_\_\_\_\_$
A. $4ab$
B. $2ab$
C. ${{a}^{2}}+2ab+{{b}^{2}}$
D. $2\left( {{a}^{2}}+{{b}^{2}} \right)$

Answer 1

Hint: Use the algebraic identity of ${{m}^{2}}-{{n}^{2}}$ to solve the given equation. So, algebraic identity of ${{m}^{2}}-{{n}^{2}}$can be given as ${{m}^{2}}-{{n}^{2}}=\left( m-n \right)\left( m+n \right)$.
After putting the identity, simplify the expression further to get the value of the given expression.

Complete step-by-step answer:

The expression in the question is ${{\left( a+b \right)}^{2}}-{\left( a-b \right)^{2}}$.
Let the value of the given expression in the problem is M. Hence, we can write ‘M’ as $M={{\left( a+b \right)}^{2}}-{\left( a-b \right)^{2}}..........\left( i \right)$
To simplify the above equation, let us suppose the value of (a + b) as ‘X’ and (a – b) as ‘Y’. Hence, we have,
a + b = X …………… (ii)
a – b = Y ………….. (iii)
Now, Replacing (a + b) and (a – b) by ‘X’ and ‘Y’ from equation (ii) and (iii) in the given expression which is written as ‘M’ in equation (i). Hence, we get,
$M={{X}^{2}}-{{Y}^{2}}.............\left( iv \right)$
Now, we can use an algebraic identity to simplify the equation (iv) further. Algebraic identity of $\left( {{m}^{2}}-{{n}^{2}} \right)$ can be used here and given as,
${{m}^{2}}-{{n}^{2}}=\left( m-n \right)\left( m+n \right)$……………… (v)
Now, we can factorize \[\left( {{X}^{2}}-{{Y}^{2}} \right)\]by using equation (v) by put m = X and n = Y. Hence, we get,
$M=\left( {{X}^{2}}-{{Y}^{2}} \right)=\left( X-Y \right)\left( X+Y \right)$
Hence, we get,
$M=\left( X-Y \right)\left( X+Y \right).........\left( vi \right)$
Now, we can get the value of M in terms of ‘a’ and ‘b’ by replacing values of ‘X’ and ‘Y’ as X = a + b and Y = a – b from the equations (ii) and (iii).
Hence, we can get equation (iv) as,
$M=\left( \left( a+b \right)-\left( a-b \right) \right)\left( \left( a+b \right)+\left( a-b \right) \right)$
Now, simplify the above equation while taking care of positive and negative signs. Hence, we get ‘M’ as,
$\begin{align}
  & M=\left( a+b-a+b \right)\left( a+b+a-b \right) \\
 & M=\left( 2b \right)\left( 2a \right)...................\left( vii \right) \\
 & M=4ab \\
\end{align}$
So, we get,
$M={{\left( a+b \right)}^{2}}-{{\left( a-b \right)}^{2}}=4ab$
So, option (A) is the correct answer.

Note: Another approach for solving this question would be that we can expand values of ${{\left( a+b \right)}^{2}}\ and\ {{\left( a-b \right)}^{2}}$by following algebraic identities:
$\begin{align}
  & {{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab \\
 & {{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab \\
\end{align}$
Now, put it in the given expression and get the answer as 4ab.
One may go wrong while putting values of x and y as (a + b) and (a – b) respectively. He/she may wrong with the step as,
\[M={{x}^{2}}-{{y}^{2}}=\left( \left( a+b \right)-\left( a-b \right) \right)\left( \left( a+b \right)+\left( a-b \right) \right)\]
Now, one may open the first bracket as (a + b – a – b) and hence will get answer ‘0’ which is wrong. It is because of the ‘ – ’ sign in front of (a – b). So, when the bracket of (a – b) opens signs of a and b will change. So, take care of it as well.