Question

Factorise: \[9{\left( {2a - b} \right)^2} - 4\left( {2a - b} \right) - 13\]

Answer 1

Hint:
Here, we have to factorize the equation. First we have to convert the equation into a quadratic form in simple form by substitution and then we have to factorize the equation. Factorization or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.

Complete step by step solution:
We are given with an equation \[9{\left( {2a - b} \right)^2} - 4\left( {2a - b} \right) - 13 = 0\]
Substituting \[2a - b = m\] in the equation, we get
\[ \Rightarrow 9{m^2} - 4m - 13 = 0\]
Now, by factoring, we get
\[ \Rightarrow 9{m^2} - 4m - 13 = 9{m^2} + 9m - 13m - 13\]
By taking out the common terms, we get
\[ \Rightarrow 9{m^2} - 4m - 13 = 9m\left( {m + 1} \right) - 13\left( {m + 1} \right)\]
Again by taking out common terms, we get
\[ \Rightarrow 9{m^2} - 4m - 13 = \left( {9m - 13} \right)\left( {m + 1} \right)\]
Let us again replace the variable \[m\] by \[2a - b\]
So, we get
\[ \Rightarrow 9{\left( {2a - b} \right)^2} - 4\left( {2a - b} \right) - 13 = \left( {9(2a - b) - 13} \right)\left( {(2a - b) + 1} \right)\]
By simplification, we get
\[ \Rightarrow 9{\left( {2a - b} \right)^2} - 4\left( {2a - b} \right) - 13 = \left( {18a - 9b - 13} \right)\left( {2a - b + 1} \right)\]

Therefore, the factorization of \[9{\left( {2a - b} \right)^2} - 4\left( {2a - b} \right) - 13\]is \[\left( {18a - 9b - 13} \right)\left( {2a - b + 1} \right)\].

Additional Information:
Factoring out common factors is applicable only If each term in the polynomial shares a common factor. The sum-product pattern is applicable only If the polynomial is of the form \[{x^2} + bx + c\], and there are factors of \[c\] that add up to \[b\]. The grouping method is applicable only If the polynomial is of the form \[a{x^2} + bx + c\], and there are factors of \[ac\] that add up to \[b\].

Note:
We know that the quadratic equation can be factorized by many ways. Factorization by grouping the terms is a method in which from the given expression a factor can be taken out from each group. Factorize each group. Now take out the factor common to group formed. These methods are more complicated since it has six terms when using the algebraic method. So, we choose the method of substitution which is easier.