Question

How do you factorize $9{{x}^{2}}-48x+64$.

Answer 1

Hint: We are given $9{{x}^{2}}-48x+64$, we are asked to find the factor of this, to do so we will first understand the type of our equation once we get that, we will take out the greatest possible term out of all the term of our expression after that we will solve further using the middle split method in which we split term and the make pair of first $2$ terms and last two terms and factorize them.

Complete step by step answer:
We are given $9{{x}^{2}}-48x+64$, we have to factorize it to the simplest form, now as we can see that the equation has the highest degree of $2$, so it means it is a quadratic equation. Now we look for the factor common to all the term in the given equation as we have $9{{x}^{2}}-48x+64$, term are $9,48,64$ they all have nothing common to all terms so we cannot factor out anything.
Now we will solve further, we will factorize the middle term.
To find the perfect pair for split of middle term for any quadratic equation $a{{x}^{2}}+bx+c$. we find $a\times c$then we find those numbers whose product is same as $a\times c$ and their sum or difference is same as $b$
Now in our equation $9{{x}^{2}}-48x+64$
We have $a=9,\,b=-48,\,c=64$
so, $a\times c=9\times 64=576$
we can see that $24\times 24=576$
and $-24+\left( -24 \right)=-48$
so we use $-24+\left( -24 \right)=-48$ to split the middle term
now we have
$9{{x}^{2}}-48x+64$= $9{{x}^{2}}+\left( -24-24 \right)x+64$opening brackets
We get,
\[9{{x}^{2}}-24-24x+64\]
Now we make pair of first two term and last two terms and try to take common terms and
\[\left( 9{{x}^{2}}-24 \right)-24x+64\]
taking common terms, we get,
$=3x\left( 3x-8 \right)-8\left( 3x-8 \right)$
As $3x-8$ is common we get, \dfrac
 $\left( 3x-8 \right)\left( 3x-8 \right)$
So we get,
$9{{x}^{2}}-48x+64$$=\left( 3x-8 \right)\left( 3x-8 \right)$
Here the factor of $9{{x}^{2}}-48x+64$ is $=\left( 3x-8 \right)\left( 3x-8 \right)$.
When we find the factor, we should know about the type of equation because the equation will tell us about the number of factors that the equation can have in the simplified form.
Like if an equation has the highest power as $5,7$ it will have $5$ term in factor term.
In our case we have the equation $9{{x}^{2}}-48x+64$. Its highest power was $2$ So we can see that we get $2$ term in factor form. Hence it means that we have reached the final term, it cannot be a factor. So, the factor form of $9{{x}^{2}}-48x+64$ is $=\left( 3x-8 \right)\left( 3x-8 \right)$.

Note: While splitting the middle term, we need to always remember the trick this says is the sign of a and c in $a{{x}^{2}}+bx+c$ are the same then b is obtained. Ways by addition of terms and is sign of a and c are opposite then b is obtained by subtraction of 2 numbers.