SOLUTION: Simplify square root of x^2+2x+1

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Question 416287: Simplify square root of x^2+2x+1
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%28x%5E2%2B2x%2B1%29
Simplifying square roots involves finding perfect square factors, if any. So to know if there are perfect square factors we have to look at the radicand (the expression inside a radical is called the radicand) in factored form. When the radicand is just a number or some variable to some power, the factors are easy to see. For multiple term radicands, like yours, you really need to do the factoring to see if there are perfect square factors.

So we start by trying to factor x%5E2+%2B+2x+%2B+1. The GCF is 1 which we rarely factor out. Since there are three terms we look next to factoring patterns or trinomial factoring. Both of these methods will work showing us that
x%5E2+%2B+2x+%2B+1+=+%28x%2B1%29%28x%2B1%29+=+%28x%2B1%29%5E2
Now that we have the radicand factored we can see that not only is there a perfect square factor but the entire radicand is a perfect square! Your expression is now:
sqrt%28%28x%2B1%29%5E2%29

The next part is actually the trickiest. One would think that
sqrt%28%28x%2B1%29%5E2%29+=+x%2B1
But this is not actually correct. Square roots are non-negative (i.e. positive or zero). Your original expression is a square root so it must be non-negative. The simplified expression must be just as non-negative as the original expression. But, without knowing what values "x" might have, we have no idea whether x+1 will be non-negative! So to ensure that our simplified expression is also non-negative we use absolute value:
sqrt%28%28x%2B1%29%5E2%29+=+abs%28x%2B1%29

Note: Sometimes problems like these include a statement like: "Assume all variables are non-negative." You did not include any statement like that in your post. If there was such a statement (or anything else that would help us know that x+1 would have to be non-negative) then the absolute value would be unnecessary.