Question

How do factor \[12{{x}^{2}}+36xy+27{{y}^{2}}\]?

Answer 1

Hint: This question is from the topic of algebra. In solving this question, we will first find the prime factorization of the term 12. After that, we will find the prime factorization of 36. After that, we will find prime factorization of 27. After solving further equation, we will use the formula of \[{{\left( a+b \right)}^{2}}\] which is equal to \[{{a}^{2}}+2\times a\times b+{{b}^{2}}\]. After using this formula and solving the further equation, we will find the factor of the term \[12{{x}^{2}}+36xy+27{{y}^{2}}\].

Complete step by step answer:
Let us solve this question.
In this question, we have asked to find the factor of the term which is given in the given in the question. The term given in the question is \[12{{x}^{2}}+36xy+27{{y}^{2}}\].
So, let us first find out the prime factorization of 12.
\[\begin{align}
  & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
Prime factorization of 12 will be \[2\times 2\times 3\times 1\].
Now, for the prime factorization of 36, we can write
\[\begin{align}
  & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
The prime factorization of 36 will be \[2\times 2\times 3\times 3\times 1\]
Now, for the prime factorization of 27, we can write
\[\begin{align}
  & 3\left| \!{\underline {\,
  27 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
So, the prime factorization of 27 will be \[3\times 3\times 3\times 1\]
Hence, we can write the term \[12{{x}^{2}}+36xy+27{{y}^{2}}\] as
\[12{{x}^{2}}+36xy+27{{y}^{2}}=2\times 2\times 3\times 1\times {{x}^{2}}+2\times 2\times 3\times 3\times 1\times x\times y+3\times 3\times 3\times 1\times {{y}^{2}}\]
Now, taking \[3\times 1\] common in the above equation, we can write
\[\Rightarrow 12{{x}^{2}}+36xy+27{{y}^{2}}=3\times 1\times \left( 2\times 2\times {{x}^{2}}+2\times 2\times 3\times x\times y+3\times 3\times {{y}^{2}} \right)\]
The above equation can also be written as
\[\Rightarrow 12{{x}^{2}}+36xy+27{{y}^{2}}=3\left( {{\left( 2x \right)}^{2}}+2\times \left( 2x \right)\times \left( 3y \right)+{{\left( 3y \right)}^{2}} \right)\]
Now, using the formula \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2\times a\times b+{{b}^{2}}\], we can write the above equation as
\[\Rightarrow 12{{x}^{2}}+36xy+27{{y}^{2}}=3{{\left( 2x-3y \right)}^{2}}\]
As we know that \[{{x}^{2}}\] can also be written as \[\left( x \right)\left( x \right)\], so we can write
\[\Rightarrow 12{{x}^{2}}+36xy+27{{y}^{2}}=3\left( 2x-3y \right)\left( 2x-3y \right)\]

Now, we have found the factor of the term \[12{{x}^{2}}+36xy+27{{y}^{2}}\]. The factor is \[3\left( 2x-3y \right)\left( 2x-3y \right)\].

Note: We should know how to find prime factorization of any number.
We should remember the following formulas for solving this type of question:
\[\left( x \right)\left( x \right)={{x}^{2}}\]
\[{{\left( a+b \right)}^{2}}={{a}^{2}}+2\times a\times b+{{b}^{2}}\]
We can solve this question by alternate method also.
The term which we have to solve is
\[12{{x}^{2}}+36xy+27{{y}^{2}}\]
As we know that 3 multiplied by 4 is 12, 3 multiplied by 12 is 36, and 3 multiplied by 9 is 27, so we can write the above term as
\[\Rightarrow 3\times 4\times {{x}^{2}}+3\times 12\times xy+3\times 9\times {{y}^{2}}\]
Now, taking 3 as common in the above term, we can write
\[\Rightarrow 3\times \left( 4\times {{x}^{2}}+12\times xy+9\times {{y}^{2}} \right)\]
Now, using the formula \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2\times a\times b+{{b}^{2}}\], we can write
\[\Rightarrow 3{{\left( 2x-3y \right)}^{2}}\]
The above term can also be written as
\[\Rightarrow 3\left( 2x-3y \right)\left( 2x-3y \right)\]