Question

Factorise: $ {a^2} + 2ab + {b^2} - 9{c^2} $

Answer 1

Hint: To find or write factored form of given equation. We first form a perfect square of first three terms of given equation by using identity and then writing so formed equation in term of $ {a^2} - {b^2} $ and simplifying or writing in factor form by using algebraic identity to get required factors of given equation.

Complete step-by-step answer:
Given equation is
 $ {a^2} + 2ab + {b^2} - 9{c^2} $
To factorise the above equation we first write the first three terms in perfect square.
First three term of given equation can be written as:
 $ {\left( a \right)^2} + 2\left( a \right)\left( b \right) + {\left( b \right)^2} - 9{c^2} $
We see that the first three terms are equivalent to expansion of $ {\left( {a + b} \right)^2} $ .
Therefore, above equation can be written as:
 $ \Rightarrow {\left( {a + b} \right)^2} - 9{c^2} $
Now, writing $ 9{c^2} $ in terms of perfect square. We have,
 $ \Rightarrow {\left( {a + b} \right)^2} - {\left( {3c} \right)^2} $
Now, using algebraic identity $ \left( {{A^2} - {B^2}} \right) = \left( {A + B} \right)\left( {A - B} \right) $ in above written equation. We have,
Taking $ A = a + b,\,\,and\,\,B = 3c $
We have,
\[\Rightarrow \left( {a + b + 3c} \right)\left( {a + b - 3c} \right)\]
Which is the required factorisation of a given equation.
So, the correct answer is “\[\left( {a + b + 3c} \right)\left( {a + b - 3c} \right)\]”.

Note: To factorise an equation there are different ways as in numeric equation we use remainder concept method but for an equation having alphabetical coefficients. We use identity or completing square methods. If terms are given such that their perfect square can be formed by using identity then use identity otherwise we can use completing square method to form it and then using it we can find required factors of given equation.