Question

How do you factor completely $4{{x}^{2}}-9{{y}^{2}}$ ?

Answer 1

Hint: To find the factor for the above equation, we need to use algebraic identity. First, rewrite the equation for applying the identity then the rewritten equation should be in the form of identity so that we can get the required factors.
The identity that we are going to use in this problem is ${{a}^{2}}-{{b}^{2}}=(a+b)(a-b)$ .

Complete step by step answer:
According to the question, we have been asked to find the factors of a two variables equation. The given equation is $4{{x}^{2}}-9{{y}^{2}}$. From this equation the square root of 4 is 2 and 9 is 3. Combine 2 and x as 2x then write that inside the brackets with the power 2. In the same way, combine 3 and y then again write 3y in a bracket with power 2.
So, the equation can be re-written as ${{(2x)}^{2}}-{{(3y)}^{2}}$ to get the perfect square.
We know the algebraic identity,
\[{{a}^{2}}-{{b}^{2}}\]= (a+b) (a-b)
According to the given equation we have a=2x and b=3y.
Since, both terms are perfect squares, we are using the above difference of squares formula.
Substitute 2x in the place of ‘a’ and 3y in the place of ‘b’.
For left hand side we get
${{a}^{2}}-{{b}^{2}}$ = ${{(2x)}^{2}}-{{(3y)}^{2}}$in the same way for the right hand side, (a+b) (a-b) = (2x+3y) (2x-3y)
So, we can get the factor as
${{(2x)}^{2}}-{{(3y)}^{2}}$= (2x+3y) (2x-3y) which is the exact form of the above algebraic identity or difference of square formula.
Hence, the required factors are $(2x+3y)(2x-3y)$.

Note: Make sure that you are using the proper algebraic Identity. Then only you can find the factors easily. A common mistake made when writing a monomial raised to a power of 2 is ${{(25x)}^{2}}=(5{{x}^{2}})$ which is incorrect. Secondly, substitute the values of ‘a ‘and ‘b’ in the correct place. Some facts are: one is a factor of every given number and every number is a factor of itself.
Every factor of a given number is either less than or equal to the given number and also it is an exact divisor. Factors multiplied together to get the given polynomial. In general, the first method for factoring polynomials will be factoring out the greatest common factor (G.C.D).