Question

How do you simplify $\dfrac{{{x}^{2}}-9}{2x+1}$?

Answer 1

Hint: In order to find the solution of the given expression that is $\dfrac{{{x}^{2}}-9}{2x+1}$ firstly we will start with converting 9 into ${{3}^{2}}$ in numerator part. After this conversion now, the numerator part has become ${{x}^{2}}-{{3}^{2}}$. In the numerator, we will apply the most common property ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$. And after applying this rule we will make necessary changes to get the required result.

Complete step by step solution:
According to the question, we have been given the expression as
$\dfrac{{{x}^{2}}-9}{2x+1}$ ……………… (1)
Now, to solve the question, we will first start with a numerator and then simplify it to get simplified value. Also, we know that denominator cannot be further simplified because it is already in linear form possible.
Now we have the numerator as ${{x}^{2}}-9$ and to simplify it further we will try to express it as ${{a}^{2}}-{{b}^{2}}$ and then we will use the property ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$ and get our final answer.
Now, we know that 9 can be written as ${{3}^{2}}$. Therefore we can say that the numerator is as same as ${{x}^{2}}-9={{x}^{2}}-{{3}^{2}}$
So we can say, equation (1) can be written as
$\Rightarrow \dfrac{{{x}^{2}}-{{3}^{2}}}{2x+1}$
According to the property ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$, we know that ${{x}^{2}}-{{3}^{2}}$ can be written as (x-3)(x+3)
Therefore, we get the numerator as ${{x}^{2}}-{{3}^{2}}=\left( x-3 \right)\left( x+3 \right)$.
And we can further write equation (1) as
$\therefore \dfrac{\left( x-3 \right)\left( x+3 \right)}{2x+1}$

Hence we can say that $\dfrac{{{x}^{2}}-9}{2x+1}$ can be further simplified as $\dfrac{\left( x-3 \right)\left( x+3 \right)}{2x+1}$.

Note: During solving the above equation we should remember the property. It is difficult to solve the problem when we have no idea about the property. We can also solve this question by adding and subtracting 3x in numerator and then taking from terms in order to find factors but that can be a lengthy method so we used the easier one.