Question

Solve the following expression, $12{{\left( 2x-3y \right)}^{2}}-16\left( 3y-2x \right)$.

Answer 1

Hint: In order to solve this question, we should have some idea about the algebraic properties like, $-\left( a-b \right)=\left( b-a \right)$. Also, we should be careful, calm and focused while solving this question as it involves multiple signs.

Complete step-by-step answer:
In this question, we have been asked to solve the given expression, that is, $12{{\left( 2x-3y \right)}^{2}}-16\left( 3y-2x \right)$. To solve this question, we should know the algebraic property that, $-\left( a-b \right)=\left( b-a \right)$. We will use this property and write the part of the given expression, $-16\left( 3y-2x \right)$ as $16\left( 2x-3y \right)$. So, applying this in the given expression, we will get,
$12{{\left( 2x-3y \right)}^{2}}+16\left( 2x-3y \right)$
We can further simplify the given expression and write it as,
$4\times 3\times \left( 2x-3y \right)\times \left( 2x-3y \right)+4\times 4\times \left( 2x-3y \right)$
We can see that in the above expression, $4\times \left( 2x-3y \right)$ is common. So, we can take this common term out. By doing so, we will get the above expression as follows,
$4\times \left( 2x-3y \right)\left( 3\left( 2x-3y \right)+4 \right)$
Now, we know that we can express the term $3\left( 2x-3y \right)$ as $\left( 6x-9y \right)$, after we open the bracket. So, we can write it in the above expression. So, we get,
$4\left( 2x-3y \right)\left( 6x-9y+4 \right)$
Hence, we can say that the given expression, $12{{\left( 2x-3y \right)}^{2}}-16\left( 3y-2x \right)$ can be expressed as $4\left( 2x-3y \right)\left( 6x-9y+4 \right)$.

Note: There is a chance of making mistakes with the use of the negative signs. So, one should be very careful while applying the signs and writing the new expressions obtained. One can also solve this question by writing ${{\left( 2x-3y \right)}^{2}}$ as ${{\left( -\left( 3y-2x \right) \right)}^{2}}$, which is the same ${{\left( 3y-2x \right)}^{2}}$ and we can take the term $4\left( 3y-2x \right)$ as common and then simplify to get the answer.