Question

Factorise the given expression: \[{{a}^{2}}-{{(b-c)}^{2}}\]
a) \[(a+b-c)(a-b+c)\]
b) \[(a-b-c)(a-b+c)\]
c) \[(a-b-c)(a-b-c)\]
d) \[(a+b-c)(a-b-c)\]

Answer 1

Hint: To solve such a type of question we will proceed by using the identity of algebraic expressions. One of the identities which is to be used is given by \[{{x}^{2}}-{{y}^{2}}=(x+y)(x-y)\].
After this we will make necessary substitutions to find the value of the given expression.

Complete Step-by-Step solution:
Given in the question is to factorise the expression \[{{a}^{2}}-{{(b-c)}^{2}}\].
To do so we will use an identity which is given as,
\[{{x}^{2}}-{{y}^{2}}=(x+y)(x-y)\]
To solve the given expression in the question we will assume the value of x as a and the value of b as (b-c).
Now using the above expression and substituting the values of x and y in the formula \[{{x}^{2}}-{{y}^{2}}=(x+y)(x-y)\] we get,
\[{{a}^{2}}-{{(b-c)}^{2}}=(a+(b-c))(a-(b-c))\]
Opening the brackets on the right-hand side of the above obtained equation we get,
\[\Rightarrow {{a}^{2}}-{{(b-c)}^{2}}=(a+b-c)(a-b+c)\]
Hence, we obtain the result as,
\[{{a}^{2}}-{{(b-c)}^{2}}=(a+b-c)(a-b+c)\], which is option (a).
Therefore, we get the answer as option (a).

Note: The possibility of errors in this question is not opening the bracket properly and not assuming one of the terms, for example, (b-c) as x or y, this leads to error in opening of bracket and problem in applying negative and positive signs which lead to incorrect Answers. Therefore, always opt for assuming a variable and then open the brackets.