Question

Factorize the following:
${{a}^{2}}-2ab+{{b}^{2}}-{{c}^{2}}$

Answer 1

Hint: To factorize the given expression we are going to use the following algebraic identity which states that: $\left( {{x}^{2}}-2xy+{{y}^{2}} \right)={{\left( x-y \right)}^{2}}$. You might be thinking why we need this algebraic identity because if you carefully look at the first three terms in the given expression, they are written in this form only. Then we will need the following algebraic identity: ${{x}^{2}}-{{y}^{2}}=\left( x-y \right)\left( x+y \right)$. Use these algebraic identities and then factorize the given expression.

Complete step by step answer:
The expression given in the above problem is as follows:
${{a}^{2}}-2ab+{{b}^{2}}-{{c}^{2}}$
If you carefully look at the first three terms of the above expression then you will find that the three terms are written in the following algebraic identity:
$\left( {{x}^{2}}-2xy+{{y}^{2}} \right)={{\left( x-y \right)}^{2}}$
Substituting $x=a,y=b$ in the above equation we get,
$\left( {{a}^{2}}-2ab+{{b}^{2}} \right)={{\left( a-b \right)}^{2}}$
Now, we are going to use the above relation in the given expression we get,
${{\left( a-b \right)}^{2}}-{{c}^{2}}$
Factoring the above expression further, we are going to use the following algebraic identity:
${{x}^{2}}-{{y}^{2}}=\left( x-y \right)\left( x+y \right)$
Substituting $x=\left( a-b \right),y=c$ in the above equation we get,
$\begin{align}
  & {{\left( a-b \right)}^{2}}-{{c}^{2}}=\left( \left( a-b \right)-c \right)\left( a-b+c \right) \\
 & \Rightarrow {{\left( a-b \right)}^{2}}-{{c}^{2}}=\left( a-b-c \right)\left( a-b+c \right) \\
\end{align}$
From the above solution, we have factorized the given expression into $\left( a-b-c \right)\left( a-b+c \right)$.
Hence, we have factorized the given expression.

Note: To solve the above problem, you need to have knowledge of the following algebraic identities:
$\left( {{x}^{2}}-2xy+{{y}^{2}} \right)={{\left( x-y \right)}^{2}}$ and ${{x}^{2}}-{{y}^{2}}=\left( x-y \right)\left( x+y \right)$. Failure of knowing these concepts will paralyze you in solving this problem so make sure you have very well understood these algebraic identities. Another thing is that you might be thinking that how we know when to use these identities in the problem, for that part, first of all, practice the simple questions related to these algebraic identities. Once you get familiarized with the identities form then your mind can easily solve these kinds of problems.