I don't understand why you have to write the absolute value sign when solving for the square root of $x$ squared.
Shouldn't the answer automatically be positive? Why is the absolute value sign needed?
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The question as stated is ambiguous. Do you mean "Why is $\sqrt{x^2}=|x|$?" or "Why is $\sqrt{x}^2=|x|$?" This is why using MathJax is good. – robjohn Jan 31 '16 at 17:42
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Long story short: because $\sqrt{(-5)^2}=5\ne -5$. – Feb 02 '16 at 00:48
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$$\sqrt{x^2}=\begin{cases}x & \text{if} \: x\geq0 \\ -x & \text{if} \: x<0 \end{cases} = |x|$$
Is it clear?
SchrodingersCat
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Downvoter, reason please? Do not go serial downvoting.. Give some reason.. – SchrodingersCat May 20 '16 at 07:43
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Yes, the square root is automatically positive - that's exactly why we need the absolute value when we write $\sqrt{x^2}=|x|$. If instead we wrote $\sqrt{x^2}=x$ that would be wrong; for $x<0$ that would say the square root of $x^2$ was not positive.
David C. Ullrich
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The answer is automatically positive, but $x$ in the radicand isn't necessarily.
Another justification:
Both are non-negative, and their squares are equal.
Bernard
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