Question

Factorise the given quadratic polynomial:- \[2{{a}^{2}}-17ab+26{{b}^{2}}\]
(a) (2a – 13b) (2a – b)
(b) (2a – b) (a – 13b)
(c) (2a – 13b) (a + 2b)
(d) (2a – 13b) (a – 2b)

Answer 1

Hint: Multiply \[2{{a}^{2}}\] and \[26{{b}^{2}}\] and use prime factorization method to write the coefficient of \[{{a}^{2}}{{b}^{2}}\] as a product of their primes. Now, split -17ab into two terms such that their product is \[52{{a}^{2}}{{b}^{2}}\] and sum is -17ab. To do this, group the factors obtained in a particular manner. Now, take the common terms together and write the given equation as a product of two terms.

Complete step-by-step solution
Here, we have been provided with the algebraic expression \[2{{a}^{2}}-17ab+26{{b}^{2}}\]. We have to factorize the given expression. So, we are going to use the method of splitting the middle term to get the answer. We will break the middle term -17ab into two terms such that their product is equal to the product is equal to the product of \[2{{a}^{2}}\] and \[26{{b}^{2}}\], and their sum is equal to -17ab. So, we have,
The product of \[2{{a}^{2}}\] and \[26{{b}^{2}}\] = \[2{{a}^{2}}\times 26{{b}^{2}}=52{{a}^{2}}{{b}^{2}}\]. Here, the co – efficient of \[{{a}^{2}}{{b}^{2}}\] is 52 so write it as a product of its prime factors, we have,
\[\Rightarrow 52=2\times 2\times 13\]
Now, we have to combine and group these factors such that their sum is -17. So, -17 can be written as: -
\[\begin{align}
  & \Rightarrow -17=-\left( 2\times 2 \right)+\left( -13 \right) \\
 & \Rightarrow -17=\left( -4 \right)+\left( -13 \right) \\
 & \Rightarrow -17ab=\left( -4ab \right)+\left( -13ab \right) \\
\end{align}\]
Therefore, we can write the given equation as: -
\[\begin{align}
  & \Rightarrow 2{{a}^{2}}-17ab+26{{b}^{2}}=2{{a}^{2}}-4ab-13ab+26{{b}^{2}} \\
 & \Rightarrow 2{{a}^{2}}-17ab+26{{b}^{2}}=2a\left( a-2b \right)-13b\left( a-2b \right) \\
\end{align}\]
Taking the common terms together, we have,
\[\Rightarrow 2{{a}^{2}}-17ab+26{{b}^{2}}=\left( a-2b \right)\left( 2a-13b \right)\]
Hence, option (d) is the correct answer.

Note: One may note that there are other methods also to solve the question. We can find the solution of the given function by assuming it to be a quadratic equation in \[{{a}^{2}}\] or \[{{b}^{2}}\]. In this way, we will get two relations between ‘a’ and ‘b’ like - ma = nb and pa = qb. Finally, write these relations as (ma – nb) (pa – qb) to get the answer. There can be a very easy method to determine the correct option, we will substitute a = b = 1 in the given equation and options and then check which of them matches. But this method can only be applied if the options are given.