Question

How do you factor $9{{x}^{2}}+4{{y}^{2}}$?

Answer 1

Hint: First write $9{{x}^{2}}$ and $4{{y}^{2}}$ in the square form of ${{\left( 3x \right)}^{2}}$ and ${{\left( 2y \right)}^{2}}$ respectively. Then try to complete the square of ${{\left( 3x+2y \right)}^{2}}$ by adding and subtracting $2\cdot 3x\cdot 2y$ as per the reverse ${{\left( a+b \right)}^{2}}$ formula i.e. ${{\left( a \right)}^{2}}+{{\left( b \right)}^{2}}+2\cdot a\cdot b={{\left( a+b \right)}^{2}}$. Do the necessary calculation to obtain the required solution, if necessary.

Complete step-by-step solution:
Completing square method: we have to convert the given expression to square terms and then factorize it by doing the necessary calculations.
Considering our expression $9{{x}^{2}}+4{{y}^{2}}$
Here, $9{{x}^{2}}$ can be written as $9{{x}^{2}}={{\left( 3x \right)}^{2}}$
Similarly, $4{{y}^{2}}$ can be written as $4{{y}^{2}}={{\left( 2y \right)}^{2}}$
Now, the expression is
$\begin{align}
  & 9{{x}^{2}}+4{{y}^{2}} \\
 & \Rightarrow {{\left( 3x \right)}^{2}}+{{\left( 2y \right)}^{2}} \\
\end{align}$
As we know ${{\left( a \right)}^{2}}+{{\left( b \right)}^{2}}+2\cdot a\cdot b={{\left( a+b \right)}^{2}}$, so to convert our expression to ${{\left( a+b \right)}^{2}}$ form we have to add and subtract the $2\cdot a\cdot b$ part.
Adding and subtracting $2\cdot 3x\cdot 2y$, we get
$\begin{align}
  & \Rightarrow {{\left( 3x \right)}^{2}}+{{\left( 2y \right)}^{2}} \\
 & \Rightarrow {{\left( 3x \right)}^{2}}+2\cdot 3x\cdot 2y+{{\left( 2y \right)}^{2}}-2\cdot 3x\cdot 2y \\
 & \Rightarrow {{\left( 3x+2y \right)}^{2}}-12xy \\
\end{align}$
This is not the factored form, but it is the maximum simplified form. Hence it is the required solution of the given question.

Note: Completing square method is the only approach for the given expression. $2\cdot a\cdot b$ part should be added and subtracted to convert it to ${{\left( a+b \right)}^{2}}$ and also to maintain the value of the expression. The given question cannot be factored completely as there is a ‘$+$ ’ sign between two terms. If there was a ‘$-$ ’ sign between the two terms of the given expression, then it could be factorized as below.
$\begin{align}
  & 9{{x}^{2}}-4{{y}^{2}} \\
 & \Rightarrow {{\left( 3x \right)}^{2}}-{{\left( 2y \right)}^{2}} \\
\end{align}$
(Here also $9{{x}^{2}}$ and $4{{y}^{2}}$ can be written as ${{\left( 3x \right)}^{2}}$ and ${{\left( 2y \right)}^{2}}$ respectively)
Now, as we know ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$
So, our expression can be further simplified as
$\Rightarrow \left( 3x-2y \right)\left( 3x+2y \right)$
This is the complete factored form.