Question

Simplify the given expression $2\left( {ab + cd} \right) - {a^2} - {b^2} + {c^2} + {d^2}$

Answer 1

Hint: To find out the solution, try to use the concept of Factorisation by Regrouping Terms. And try to find out the common factors in each group.

Complete step-by-step answer:

As per the question we have to simplify the given equation, which is
$2\left( {ab + cd} \right) - {a^2} - {b^2} + {c^2} + {d^2}$
$ = \left( {2ab + 2cd} \right) - {a^2} - {b^2} + {c^2} + {d^2}$
$ = \left( {{c^2} + 2cd + {d^2}} \right) - \left( {{a^2} - 2ab + {b^2}} \right)$$..eq\left( i \right)$

If we observe the $eq\left( i \right)$ we will be found out they are following the identities

First case,\[{\left( {a + b} \right)^2}\] and Second case,${\left( {a - b} \right)^2}$

Now we will use these identities in the question,
$ = \left( {{c^2} + 2cd + {d^2}} \right) - \left( {{a^2} - 2ab + {b^2}} \right)$$..eq\left( i \right)$
$ = {\left( {c + d} \right)^2} - {\left( {a - b} \right)^2}$

And the above expression is nothing but the algebraic identity \[{a^2} - {b^2}\]
$ = \left( {c + d + a - b} \right)\left( {c + d - a + b} \right)$

Note: In these questions, we will attempt it by the Grouping Terms for Factorization can be done only when after taking out common factors in each group there should be a common term in each group. We will take the common factors and try to find out the common terms in each group.