Question

Factorize using appropriate identities.
$4{y^2} - 4y + 1$

Answer 1

Hint: Here, we are required to factorize a given expression using appropriate identities. First of all, we will identify the nature of the given expression whether it is a linear, quadratic, cubic or any other expression. Then, we will use the appropriate identity to break the expression in that form. Finally, we will apply the identity to get the required factors of the given expression.

Formula Used: We will use the formula ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$.

Complete step-by-step answer:
A quadratic equation is an equation having an integral non-negative power and it can always be represented in the form of $a{x^2} + bx + c$.
 Now, in this question, this given expression is known as a quadratic expression because it is of the form $a{x^2} + bx + c$ and it is having a power 2. Also, if we try to find the roots of this expression, then we will be able to find two roots only.
Given quadratic expression is:
$4{y^2} - 4y + 1$
Now, we can write it in the form of:
$ \Rightarrow 4{y^2} - 4y + 1 = {\left( {2y} \right)^2} - 2\left( {2y} \right)\left( 1 \right) + {\left( 1 \right)^2}$………………………………….$\left( 1 \right)$
Also, if we try to compare this with the given expression, then we can see that it is the same expression and only the way of writing it has changed.
Now, the equation $\left( 1 \right)$ is of the form ${a^2} - 2ab + {b^2}$, so we can use the identity,
${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$.
Hence, comparing this with equation $\left( 1 \right)$ and substituting $2y = a$ and $1 = b$ in the above formula, we get,
$ \Rightarrow 4{y^2} - 4y + 1 = {\left( {2y - 1} \right)^2}$
Since, the bracket has a power 2, i.e. square, hence, we can also write this as:
\[ \Rightarrow 4{y^2} - 4y + 1 = \left( {2y - 1} \right)\left( {2y - 1} \right)\]
Therefore, the factorization of $4{y^2} - 4y + 1$ is (2y - 1)(2y - 1)
Hence, this is the required answer.

Note: Since, in this question, it was mentioned to use identities to solve this question hence, we have used this identity. Otherwise, we can also solve this question using the middle term split as below:
Given expression is $4{y^2} - 4y + 1$.
Now, splitting the middle term, we get
$ \Rightarrow 4{y^2} - 4y + 1 = 4{y^2} - 2y - 2y + 1$
$ \Rightarrow 4{y^2} - 4y + 1 = 2y\left( {2y - 1} \right) - 1\left( {2y - 1} \right)$
Factoring out common terms, we get
$ \Rightarrow 4{y^2} - 4y + 1 = \left( {2y - 1} \right)\left( {2y - 1} \right)$
Therefore, we can also use middle term splitting for factoring a given quadratic expression if it is not mentioned to use the identities