Question

Factorize the given expression, \[{{x}^{2}}+4{{y}^{2}}-9{{z}^{2}}-4xy\].
a.\[[\left( x-4y-3z \right)\left( x-4y+3z \right)]\]
b.\[[\left( x-2y-3z \right)\left( x-2y+3z \right)]\]
c.\[[\left( x+2y-3z \right)\left( x+2y+3z \right)]\]
d.None

Answer 1

Hint: Expand the given expression using basic identities. Substitute the identities and formulate the expression by factoring it.

Complete step-by-step answer:
We have been given the expression, \[{{x}^{2}}+4{{y}^{2}}-9{{z}^{2}}-4xy\].

Let us rearrange them as, \[\left( {{x}^{2}}+4{{y}^{2}}-4xy \right)-9{{z}^{2}}\].

Now, \[\left( {{x}^{2}}+4{{y}^{2}}-4xy \right)\] is similar to the expansion of \[{{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\].
\[\begin{align}
  & =\left[ {{x}^{2}}+{{\left( 2y \right)}^{2}}-2xy \right]-9{{z}^{2}} \\
 & ={{\left( x-2y \right)}^{2}}-9{{z}^{2}} \\
 & ={{\left( x-2y \right)}^{2}}-{{\left( 3z \right)}^{2}} \\
\end{align}\]

Now this above expression is of the form \[{{a}^{2}}-{{b}^{2}}\]. Thus, we can write it as \[\left( a-b \right)\left( a+b \right)\].
\[\therefore {{\left( x-2y \right)}^{2}}-{{\left( 3z \right)}^{2}}=\left( x-2y-3z \right)\left( x-2y+3z \right)\].

Thus, by factorization we got, \[{{x}^{2}}+4{{y}^{2}}-9{{z}^{2}}-4xy=\left( x-2y-3z \right)\left( x-2y+3z \right)\].
\[\therefore \]Option (b) is the correct answer.

Note: Don’t take and express the identities together for \[{{x}^{2}},{{y}^{2}}\] and \[{{z}^{2}}\]. Study the question and rearrange it to get the required similarity to the basic identity. Substitute and factorize the expression to required form.