Question

: How do you simplify: square root of \[{{x}^{2}}-4x+4\]?

Answer 1

Hint: For the given problem we are given to solve the square root of \[{{x}^{2}}-4x+4\]. First of all we have to factorize the equation \[{{x}^{2}}-4x+4\] or simplify the equation. After factoring the equation find the square root for the factored equation.

Complete step by step answer:
For the given problem, we are given to solve the square root of \[{{x}^{2}}-4x+4\].
For finding the square root of the equation \[{{x}^{2}}-4x+4\]. We have to consider this as equation (1).
Let us assume the equation as ‘S’ and consider it as equation (1).
\[S={{x}^{2}}-4x+4...........\left( 1 \right)\]
Now we have to factorize the equation to get the square root of equation (1).
As we know the formula\[{{\left( x-a \right)}^{2}}={{x}^{2}}-2.a.b+{{a}^{2}}\], let us consider this formula as formula ($f_{1}$).
\[{{\left( x-a \right)}^{2}}={{x}^{2}}-2.a.b-{{a}^{2}}.........\left( f_1 \right)\]
Let us rewrite the term -4x as -2.2.x, we get
\[\Rightarrow S={{x}^{2}}-2.2.x+4\]
Let us consider the above equation as equation (2).
\[S={{x}^{2}}-2.2.x+4.........\left( 2 \right)\]
By comparing the equation (2) with formula ($f_{1}$), we can say the term ‘a’ in equation (2) is ‘2’.
So therefore we can rewrite the equation (2) as
\[S={{\left( x-2 \right)}^{2}}\]
Let us consider the equation above equation as equation (3).
\[S={{\left( x-2 \right)}^{2}}...........\left( 3 \right)\]
Now let us do square root for the equation (3), we get
\[\begin{align}
  & \Rightarrow S=\sqrt{{{\left( x-2 \right)}^{2}}} \\
 & \Rightarrow S=+\left( x-2 \right)\text{or-}\left( x-2 \right) \\
\end{align}\]
Our solution consists two roots namely (x-2) and –(x-2)
Since we use \[\sqrt{{}}\] to denote the positive square root, we need to pick whichever is non-negative of \[\left( x-2 \right)\]and\[-\left( x-2 \right)\].
We can express that as\[\left| x-2 \right|\].

Note: We should note a point that the square root of any equation gives two equations, where one is positive and negative. We can only find square roots for perfect squared equations but not for all equations. Students should compare the equation with the correct formula to get the correct answer.