Question

Factorize: \[6xy - 4y + 6 - 9x\]

Answer 1

Hint:
Here we will simplify the equation by taking the common from the first two terms and the last two terms of the given equation. Then we will form the equation in terms of the factors as factors are the smallest numbers with which the given number is divisible and their multiplication will give the original number.

Complete step by step solution:
Given equation is \[6xy - 4y + 6 - 9x\].
Firstly we will take the maximum common from the first two terms of the given equation i.e. \[2y\] from the first two terms. Therefore, we get
\[ \Rightarrow 2y\left( {3x - 2} \right) + 6 - 9x\]
Now we will take the maximum common from the last two terms of the equation i.e. \[ - 3\] from the last two terms. Therefore, we get
\[ \Rightarrow 2y\left( {3x - 2} \right) - 3\left( {3x - 2} \right)\]
Now we will take \[\left( {3x - 2} \right)\] common from both the terms of the above equation to get the final fectors of the given equation. Therefore, we get
\[ \Rightarrow \left( {3x - 2} \right)\left( {2y - 3} \right)\]

Hence, \[\left( {3x - 2} \right)\& \left( {2y - 3} \right)\] are the two factors of the given equation i.e. \[6xy - 4y + 6 - 9x\].

Note:
We should know that the process of the formation of the factors is generally known as factorization. Factorization is the process in which a number is written in the forms of its small factors which on multiplication give the original number. Here we have to note that in such types of questions the basic algebraic identities may get used to solve and find the factors of the equation. Algebraic identities are equations where the value of the left-hand side of the equation is identically equal to the value of the right-hand side of the equation. Generally in these types of questions algebraic identities are used to solve. It is very important that we learn about algebraic identities in mathematics.
\[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
\[{\left( {a - b} \right)^2}\; = {a^2} - 2ab + {b^2}\]
\[{a^2} - {b^2} = (a - b)(a + b)\]