Question

How do you factor \[{{a}^{2}}-4ab+4{{b}^{2}}\]?

Answer 1

Hint: Consider ‘a’ as the variable and ‘b’ as the constant and assume the given equation as a quadratic equation in ‘a’. Now, use the middle term split method to factorize \[{{a}^{2}}-4ab+4{{b}^{2}}\]. Split -4b in such a way, into two terms, that their sum is -4b and the product is \[4{{b}^{2}}\]. For this process, find the prime factors of 4 and combine them in such a way so that we can get our conditions satisfied. Finally, take common terms together and write \[{{a}^{2}}-4ab+4{{b}^{2}}\] as a product of two terms.

Complete step by step answer:
Here, we have been asked to factorize \[{{a}^{2}}-4ab+4{{b}^{2}}\]. Here, we can consider ‘b’ as constant and ‘a’ as variables so that the given equation becomes a quadratic equation in ‘a’.
Now, let us use the middle term split method for the factorization. It says that we have to split the middle term which is -4b into two terms such that their sum is -4b and the product is \[4{{b}^{2}}\]. To do this, first, we need to find all the prime factors of 4. So, let us find it.
We know that 4 can be written as: - \[4=2\times 2\] as the product of its primes. Now, we have to group these factors in such a way that our conditions of the middle term split method is satisfied. So, we have,
(i) \[\left( -2b \right)+\left( -2b \right)=-4b\]
(ii) \[\left( -2b \right)\times \left( -2b \right)=4{{b}^{2}}\]
Hence, both the conditions of the middle term split method are satisfied. So, the quadratic polynomial can be written as: -
\[\begin{align}
  & \Rightarrow {{a}^{2}}-4ab+4{{b}^{2}}={{a}^{2}}-2ab-2ab+4{{b}^{2}} \\
 & \Rightarrow {{a}^{2}}-4ab+4{{b}^{2}}=a\left( a-2b \right)-2b\left( a-2b \right) \\
\end{align}\]
Taking \[\left( a-2b \right)\] common in the R.H.S., we get,
\[\Rightarrow {{a}^{2}}-4ab+4{{b}^{2}}=\left( a-2b \right)\left( a-2b \right)={{\left( a-2b \right)}^{2}}\]
Hence, \[{{\left( a-2b \right)}^{2}}\] is the factored form of the given quadratic polynomial.

Note:
One may note that here we have considered ‘a’ as the variable and ‘b’ as the constant. You can also consider ‘b’ as the variable and ‘a’ as the constant and rearrange the equation as \[4{{b}^{2}}-4ab+{{a}^{2}}\] and then apply the middle term split method. It will not alter the answer. You can also write the given equation as: - \[{{a}^{2}}-2.a.2b+{{\left( 2b \right)}^{2}}\] and directly apply the identity: - \[{{x}^{2}}-2xy+{{y}^{2}}={{\left( x-y \right)}^{2}}\] to get the answer.