Question

How do you write $9{{x}^{2}}-64$ in factored form?

Answer 1

Hint: In the above problem, we are asked to write the expression $9{{x}^{2}}-64$ in factored form. For that, you can see that we can write 9 as ${{3}^{2}}$ and 64 as ${{8}^{2}}$ then the above expression will look as ${{\left( 3x \right)}^{2}}-{{8}^{2}}$. Now, this expression is of the form of the algebraic expression ${{a}^{2}}-{{b}^{2}}$ and we know there is an identity of this algebraic expression which is given as: ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$ so using this identity we can factorize the given expression.

Complete step by step solution:
In the above problem, the expression given above which we have to factorize is as follows:
$9{{x}^{2}}-64$
Now, as you can see that 9 and 64 written in the above expression is the perfect square of 3 and 8. So, writing 9 and 64 as ${{3}^{2}}$ and ${{8}^{2}}$ respectively we get,
$\begin{align}
  & {{3}^{2}}{{x}^{2}}-{{8}^{2}} \\
 & ={{\left( 3x \right)}^{2}}-{{8}^{2}} \\
\end{align}$
The above expression is in the form of algebraic expression ${{a}^{2}}-{{b}^{2}}$ and we know there is an identity of this algebraic expression which is equal to:
${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$
Now, substituting $a=3x$ and $b=8$ in the above equation we get,
${{\left( 3x \right)}^{2}}-{{8}^{2}}=\left( 3x-8 \right)\left( 3x+8 \right)$
From the above, we have factorized the given expression to $\left( 3x-8 \right)\left( 3x+8 \right)$.
Hence, the factored form of $9{{x}^{2}}-64$ is equal to $\left( 3x-8 \right)\left( 3x+8 \right)$.

Note: You can check the factors which you got above are correct or not by multiplying the two factors and see whether you are getting the same expression which is given in the above problem.
The factored form which we got above is as follows:
$\left( 3x-8 \right)\left( 3x+8 \right)$
Multiplying 3x by $\left( 3x+8 \right)$ and then multiplying 8 by $\left( 3x+8 \right)$ we get,
$\begin{align}
  & 3x\left( 3x+8 \right)-8\left( 3x+8 \right) \\
 & =3x\left( 3x \right)+3x\left( 8 \right)-24x-64 \\
 & =9{{x}^{2}}+24x-24x-64 \\
\end{align}$
In the above expression, $24x$ will be cancelled out and we are left with:
$9{{x}^{2}}-64$
Hence, we have got the same expression which we have started with.