Question

How do you factor $9{{x}^{2}}-121$ ?

Answer 1

Hint: Problems of factorisation of this type can easily be done by writing ${{3}^{2}}$ instead of $9$ and ${{11}^{2}}$ instead of $121$ . The expression is simplified into ${{\left( 3x \right)}^{2}}-{{11}^{2}}$. After that we factorize ${{\left( 3x \right)}^{2}}-{{11}^{2}}$ using the formula of subtraction of two squares which is
${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$
So, ${{\left( 3x \right)}^{2}}-{{11}^{2}}$ becomes $\left( 3x+11 \right)\left( 3x-11 \right)$ which gets us the final answer.

Complete step by step answer:
The given expression we have is
$9{{x}^{2}}-121$
As, we can write ${{3}^{2}}$ instead of $9$ and ${{11}^{2}}$ instead of $121$ and write the above expression as
$\Rightarrow {{3}^{2}}\cdot {{x}^{2}}-{{11}^{2}}$
Which can be further simplified by writing $3$ and $x$ inside one bracket and the expression thus becomes
$\Rightarrow {{\left( 3x \right)}^{2}}-{{11}^{2}}....\text{expression}1$
The above quadratic expression can be compared with the expression of subtraction of two squares, which is ${{a}^{2}}-{{b}^{2}}$. This can be factorized as
${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)....\text{expression}2$
Comparing between the $\text{expression}1$ and $\text{expression}2$ we get,
$a=3x$ and $b=11$
Factorising the $\text{expression}1$ accordingly, we get
$\Rightarrow \left( 3x+11 \right)\left( 3x-11 \right)$
Therefore, we can conclude that the given expression $9{{x}^{2}}-121$ can be factorized as $\left( 3x+11 \right)\left( 3x-11 \right)$ .

Note: We might get confused by seeing a quadratic term and a linear term together but all the terms can be easily simplified and converted into the same power, which in this case is $2$ . The other method we can use to solve this problem is the vanishing factor method. In that case we can see that the highest power $x$ have is $2$ . Hence, if we take the above expression as an equation, we will have two solutions of $x$ . One of the values is $\dfrac{11}{3}$ . This means that $\left( 3x-11 \right)$ is a factor of the given expression. Now, dividing the given expression by $\left( 3x-11 \right)$gives the other factor.