Question

Factorise: $ 20{a^2} - 45{b^2} $

Answer 1

Hint: In this type of question solve the expression by taking 5 as common from both terms and then by using the identity of $ {x^2} - {y^2} $ . This identity is applicable for the equation having subtraction between the values which are perfect squares.

Complete step-by-step answer:
The given expression for factorisation is $ 20{a^2} - 45{b^2} $ .
Here we see that 20 and 45 are the multiple of 5. If we divide the expression from 5 then the remaining expression has only perfect squares.
So, dividing the whole expression by 5 then we get,
 $
 \Rightarrow \dfrac{{20{a^2}}}{5} - \dfrac{{45{b^2}}}{5}\\
 \Rightarrow 4{a^2} - 9{b^2}
 $
The given expression has the degree of 2, as the highest exponent in the expression is 2. Now we check the end terms of the expression which are $ 4{a^2} $ and $ 9{b^2} $ .
Here, we found that $ 4{a^2} $ is the perfect square of 2a but $ 9{b^2} $ is a perfect square of $ 3b $ , hence by using the identity of $ {x^2} - {y^2} $ to find the factor of the given expression.
Hence the expression can be written as:
 $ \Rightarrow {\left( {2a} \right)^2} - {\left( {3b} \right)^2} $
So, as we know that,
  $ {x^2} - {y^2} = \left( {x + y} \right)\left( {x - y} \right) $
Comparing the given expression with the identity then we will get,
 $ x = 2a,y = 3b $
Substituting the values of x and y in the identity then we will get,
 $
{\left( {2a} \right)^2} - {\left( {3b} \right)^2} = \left( {2a + 3b} \right)\left( {2a - 3b} \right)\\
4{a^2} - 9{b^2} = \left( {2a + 3b} \right)\left( {2a - 3b} \right)
 $
Now, multiply 5 to both sides in the above equation then we get,
 $
5\left( {4{a^2} - 9{b^2}} \right) = 5\left( {2a + 3b} \right)\left( {2a - 3b} \right)\\
20{a^2} - 45{b^2} = 5\left( {2a + 3b} \right)\left( {2a - 3b} \right)
 $
Therefore, $ 5\left( {2a + 3b} \right)\left( {2a - 3b} \right) $ is the required factor of the given expression $ 20{a^2} - 45{b^2} $ and $ 20{a^2} - 45{b^2} $ completely divisible by these factors.

Note: This expression can also be factorise by using the middle term splitting method and then carry out common from the given expression. Here, $ {x^2} - {y^2} $ identity is applicable on the equation having only perfect squares.