Question

Find the factorization of equation $4{x^2} - 4x + 1 = 0$ ?

Answer 1

Hint: We have given a quadratic equation of the form \[a{x^2} + bx + c = 0\] is provided to us. We will split the middle term and then take the common terms from the first two terms and last two terms individually to identify its components. Then we will get the factored form.

Complete step by step answer:
We have the equation $4{x^2} - 4x + 1 = 0$ and it is a quadratic equation of the form \[a{x^2} + bx + c = 0\]
$ \Rightarrow 4{x^2} - 4x + 1 = 0$
We will use mid-term split formula
$ \Rightarrow 4{x^2} - 2x - 2x + 1 = 0$
We have taken 2x and -1 common from the first two terms and last two terms respectively.
$ \Rightarrow 2x(2x - 1) - 1(2x - 1) = 0$
We have rewritten the expression in factors form
$ \Rightarrow (2x - 1) \times (2x - 1) = 0$
Since, the product of two factors is zero, so
$ \Rightarrow (2x - 1) = 0$
$ \Rightarrow x = \dfrac{1}{2}$
The factorization of $4{x^2} - 4x + 1 = 0$ is $(2x - 1) \times (2x - 1) = 0$
Hence, the factorization of $4{x^2} - 4x + 1 = 0$ is \[{(2x - 1)^2} = 0\]

Note:
We should note that discriminants will tell us about the nature of roots.
1. If the value of D > 0 then the roots will be real and distinct.
2. If the value of D = 0 then the roots will be real and equal.
3. If the value of D < 0 then the roots will be complex and conjugate.