Question

Factorize the following : \[80{{a}^{2}}-45{{b}^{2}}\].

Answer 1

Hint:As we know that the factorization means to write the given expression as a product of its factors. We will take the common from the each terms of the expression then we will use the identity given as follows:
\[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\]

Complete step-by-step answer:
We have been given the expression as following:
\[80{{a}^{2}}-45{{b}^{2}}\]
Taking 5 as a common from each term of the expression, we get,
$5\left( 16{{a}^{2}}-9{{b}^{2}} \right) $
We can write above equation as,
$5\left[ {{\left( 4a \right)}^{2}}-{{\left( 3b \right)}^{2}} \right] $
Here we will use the identity \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\].
Where, $a=4a+3b$ , $b=4a-3b$
\[=5\left( 4a+3b \right)\left( 4a-3b \right)\]
Hence the factorization of the given expression is \[5\left( 4a+3b \right)\left( 4a-3b \right)\]

Note: Just remember that in factorization of any expression first try to arrange them according to the degree of the variables then take common from each term of the expression if possible.
Also, factorization of an algebraic expression seems tricky, but with knowledge of identities, it becomes easy and fast. So try to remember the different basic identities of algebra.
Also, remember that some equality is true for every value of the variable in an expression is called identity. While factorizing an algebraic expression we use these to reach an irreducible form of the expression.