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Question: What is the square root of \({{x}^{2}}+4x+4\)?...

What is the square root of x2+4x+4{{x}^{2}}+4x+4?

Explanation

Solution

For solving this question you should know about the square root of expressions. In this problem we will take the square root of this expression but it is not in a form of parts. So, first we will divide it in parts and then the parts will make a term which will be whole square and then we will take it under root.

Complete step-by-step solution:
According to our question we have to find the square root of x2+4x+4{{x}^{2}}+4x+4. So, as we know that the square root of any term will be the same term whose squared value is given for taking the square root. And if any terms are given in any question, then we always try to take that as a full square form and if it becomes like it then we directly take square root of that, but if the given expression is not in a squared form, then we try to make it in a squared form and then we add or subtract any term which will make it a squared form. And then we take it’s square root. But this makes an extra term in that. So, we can only make a squared form from this method but there will be an extra term along with it. So, it will not be possible to find the square root. One more method is to take a common factor and then divide the rest of the expression which gives us roots for that but does not give a square root.
So, according to the question, our expression is x2+4x+4{{x}^{2}}+4x+4.
Now divide 4x4x as 2x+2x2x+2x,
⇒(x2+2x)+(2x+4)\Rightarrow \left( {{x}^{2}}+2x \right)+\left( 2x+4 \right)
Now take xx common from first term and 2 from second term,
⇒x(x+2)+2(x+2)\Rightarrow x\left( x+2 \right)+2\left( x+2 \right)
Mow take (x+2)\left( x+2 \right) common from whole expression,
=(x+2)(x+2) =(x+2)2 \begin{aligned} & =\left( x+2 \right)\left( x+2 \right) \\\ & ={{\left( x+2 \right)}^{2}} \\\ \end{aligned}
Now take the square root of this,
(x+2)2=x+2\sqrt{{{\left( x+2 \right)}^{2}}}=x+2
So, the square root is x+2x+2.

Note: In this type of questions always try to make a whole square of a term because if this will not be made, then it is not possible to make a square root of that term. And if we are asked to find only roots then we can find them by different methods.