Question

How do you factor completely $16{{x}^{2}}-25{{y}^{2}}$ ?

Answer 1

Hint: To get the factor of a given equation that is $16{{x}^{2}}-25{{y}^{2}}$ , we will make every term of the equation of the given question as a perfect square. After that the question will show a difference of perfect squares. So, we will use the formula of difference of perfect square that is:
$\Rightarrow \left( {{a}^{2}}-{{b}^{2}} \right)=\left( a-b \right)\left( a+b \right)$

Complete step by step solution:
Since, we have the equation of the question as:
$\Rightarrow 16{{x}^{2}}-25{{y}^{2}}$
Since, we know that both the numbers $16$ and $25$ are perfect square numbers where $16$ is square of $4$ and $25$ is square of $5$ . Here, we can see that:
$\Rightarrow {{4}^{2}}=4\times 4=16$
And
$\Rightarrow {{5}^{2}}=5\times 5=25$
So, we can write the equation of the question as:
$\Rightarrow {{4}^{2}}\times {{x}^{2}}-{{5}^{2}}\times {{y}^{2}}$
Now, we will make the term of the above equation a perfect square as:
$\Rightarrow {{\left( 4x \right)}^{2}}-{{\left( 5y \right)}^{2}}$
Now, we got the given equation of the question in the form of the difference of perfect square, we will use the following formula:
$\Rightarrow \left( {{a}^{2}}-{{b}^{2}} \right)=\left( a-b \right)\left( a+b \right)$
After using the above formula, we will get the equation of question as:
$\Rightarrow \left( 4x-5y \right)\left( 4x+5y \right)$
Now, we cannot factorize the above equation further. Hence, This is the factorization of the question.

Note: Here, we can check whether the solution of the question is correct or not in the following way as:
Since, we have the factor of the given question as:
$\Rightarrow \left( 4x-5y \right)\left( 4x+5y \right)$
Now, we will open one bracket of the above equation as:
$\Rightarrow 4x\left( 4x+5y \right)-5y\left( 4x+5y \right)$
Here, we will multiply them as:
$\Rightarrow 4x\times 4x+4x\times 5y-5y\times 4x-5y\times 5y$
$\Rightarrow 16{{x}^{2}}+20xy-20xy-25{{y}^{2}}$
After eliminating the term of $20xy$ because of opposite sign as:
$\Rightarrow 16{{x}^{2}}-25{{y}^{2}}$
Thus we got the given question. Hence, the solution of the question is correct.