Question

How do you factor $36{{x}^{2}}-60x+25$?

Answer 1

Hint: The expression given above is the quadratic expression because the degree of this expression is 2. We are going to factorize the given quadratic expression by multiplying the coefficient of ${{x}^{2}}$ and the constant. Then we will find the factors of this multiplication and we will add or subtract these factors in such a manner so that we will get the coefficient of x and then replace the coefficient of x with these factors. And then it’s easy to factorize.

Complete step-by-step answer:
The expression given in the above problem is as follows:
$36{{x}^{2}}-60x+25$
The coefficient of ${{x}^{2}}$ is 36 in the above and constant is 25 so multiplying 36 by 25 we get 900 so factoring 900 we get,
Factorization of 900 is as follows:
$\begin{align}
  & 900=1\times 900 \\
 & 900=2\times 450 \\
 & 900=3\times 300 \\
 & 900=4\times 225 \\
 & 900=5\times 180 \\
 & 900=6\times 150 \\
 & 900=9\times 10 \\
 & 900=30\times 30 \\
\end{align}$
There can be more factors are possible but we have shown some of the factors and if you look at the last factors then adding these two factors we get 60 so substituting (30 + 30) in place of 60 in the above expression we get,
$\begin{align}
  & \Rightarrow 36{{x}^{2}}-\left( 30+30 \right)x+25 \\
 & \Rightarrow 36{{x}^{2}}-30x-30x+25 \\
\end{align}$
Taking $6x$ as common from the first two terms and -5 from the last two terms we get,
$\Rightarrow 6x\left( 6x-5 \right)-5\left( 6x-5 \right)$
Now, taking $6x-5$ as common from the above expression we get,
$\begin{align}
  & \Rightarrow \left( 6x-5 \right)\left( 6x-5 \right) \\
 & ={{\left( 6x-5 \right)}^{2}} \\
\end{align}$
Hence, we have factorized the given quadratic expression and the factors are ${{\left( 6x-5 \right)}^{2}}$.

Note: The alternative approach to the above problem is as follows:
The expression given in the above problem is as follows:
$36{{x}^{2}}-60x+25$
If you look carefully at the first and the last term then you will find that first and the last term both are perfect squares.
${{\left( 6x \right)}^{2}}-60x+{{\left( 5 \right)}^{2}}$
Now, we can make the above expression as ${{a}^{2}}-2ab+{{b}^{2}}={{\left( a-b \right)}^{2}}$ by writing $60=2\times 6x\times 5$ in the above expression we get,
$\begin{align}
  & \Rightarrow {{\left( 6x \right)}^{2}}-2\left( 6x \right)\left( 5 \right)+{{\left( 5 \right)}^{2}} \\
 & ={{\left( 6x-5 \right)}^{2}} \\
\end{align}$