Question

How do you factor completely $9{{x}^{2}}-25{{y}^{2}}$?

Answer 1

Hint: Now note that the given expression is a degree 2 polynomial in two variables. To simplify the expression we will first write $9{{x}^{2}}={{\left( 3x \right)}^{2}}$ and $25{{y}^{2}}={{\left( 5y \right)}^{2}}$ . Now we know that ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$ . Hence using this in the obtained expression we will get the factors of the expression.

Complete step by step solution:
Now we are given with a degree 2 polynomial in two variables x and y.
We want to factorize the whole expression.
Now to factorize means to write the expression in terms of its factors.
Now we want to find linear factors for the whole expression.
Now consider the given expression $9{{x}^{2}}-25{{y}^{2}}$
Now we know that $9={{3}^{2}}$ and $25={{5}^{2}}$ . Hence we will substitute these values in the expression to get,
$\Rightarrow {{3}^{2}}{{x}^{2}}-{{5}^{2}}{{y}^{2}}$
Now we know for any numbers a and b we have ${{a}^{m}}{{b}^{m}}={{\left( ab \right)}^{m}}$ . Hence we will use this property of indices in the above expression. Using this we get,
Now we will use the above formula where a = 3x, b = 5y and m = 2.
$\Rightarrow {{\left( 3x \right)}^{2}}-{{\left( 5y \right)}^{2}}$
Now the above expression is in the form ${{a}^{2}}-{{b}^{2}}$ and we know that the expansion of ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$ . Hence we have,
$\Rightarrow \left( 3x-5y \right)\left( 3x+5y \right)$
Hence we have $9{{x}^{2}}-25{{y}^{2}}=\left( 3x-5y \right)\left( 3x+5y \right)$
Now the given expression is factored and the factors of the expression are $\left( 3x-5y \right)$ and $\left( 3x+5y \right)$ .

Note: Now note that we can also simplify the expression by adding and subtracting the term $\left( 3x \right)\left( 5y \right)=15xy$ . Hence we get the expression as $9{{x}^{2}}+15xy-15xy-25{{y}^{2}}$. Now taking 3x common from the first two terms and -5y common from the last two terms we can simplify the obtained expression. Hence we will get the factors of the expression.