Question

Simplify the following expression $4{{a}^{2}}-9{{b}^{2}}-2a-3b$

Answer 1

Hint: To solve this question we need to know the concept of expansion and how the term could be expanded and written in short form so that the further calculation becomes easy. We start solving by analysing the problem. We find the square root of coefficient of $''a''$ and $''b''$. We use the formula ${{p}^{2}}-{{q}^{2}}=\left( p-q \right)\left( p+q \right)$ to solve the question.

Complete step by step solution:
The question asks us to solve the equation $4{{a}^{2}}-9{{b}^{2}}-2a-3b$ which is given. On analysing the given equation $4{{a}^{2}}-9{{b}^{2}}-2a-3b$, we see that which is four terms separated either by addition or subtraction. We see that there are two terms having variables in terms of $''a''$ and $''b''$.
It is very much visible from the question that the coefficients of $''a''$ and $''b''$ are also the square of a certain natural number. To solve the equation we need to find the square root of the coefficients which are $4$ and $9$ respectively. On finding the square root we get:
$\Rightarrow \sqrt{4}$
$\Rightarrow \sqrt{2\times 2}$
$\Rightarrow 2$
Similarly for the coefficient of $''b''$ which is $9$, we get:
$\Rightarrow \sqrt{9}$
$\Rightarrow \sqrt{3\times 3}$
$\Rightarrow 3$
Now the given question $4{{a}^{2}}-9{{b}^{2}}-2a-3b$could be written as:
$\Rightarrow {{\left( 2a \right)}^{2}}-{{\left( 3b \right)}^{2}}-2a-3b$
To calculate the further we need to use the formula ${{p}^{2}}-{{q}^{2}}=\left( p-q \right)\left( p+q \right)$ . On applying the formula in the above equation we get:
$\Rightarrow \left( 2a-3b \right)\left( 2a+3b \right)-\left( 2a+3b \right)$
We can see $\left( 2a+3b \right)$ is common, so taking it outside:
$\Rightarrow \left( 2a+3b \right)\left( 2a-3b-1 \right)$
$\therefore $On solving $4{{a}^{2}}-9{{b}^{2}}-2a-3b$ we get \[\left( 2a+3b \right)\left( 2a-3b-1 \right)\].

Note: The problem could be rechecked by calculating the terms which means by multiplying the terms. On doing this we get:
 \[\Rightarrow \left( 2a+3b \right)\left( 2a-3b-1 \right)\]
\[\Rightarrow 2a\left( 2a-3b-1 \right)+3b\left( 2a-3b-1 \right)\]
\[\Rightarrow 4{{a}^{2}}-6ab-2a+6ab-9{{b}^{2}}-3b\]
This above expansion contain $6ab$ with both positive and negative sign so this term get cancelled, resulting in:
$\Rightarrow 4{{a}^{2}}-9{{b}^{2}}-2a-3b$
Since the answer after multiplication is the same as the question, so the answer we got is correct.