You can't have the square root of a negative number, because when you multiply two numbers of the same sign, you cannot get a negative number. (sqrt) -1 is an imaginary number symbolized by i.
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Every positive real number has two square roots: one negative and one positive. As a result, the square root mapping is one-to-many and so is not a mathematical function. One way to make it a function is to restrict the range to non-negative real numbers. These are the non-negative square roots.
If a is any number, then a squared = (-a) squared, so one might say that a and -a are both square roots of a squared. However, the square root symbol always means the positive square root.
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If you square an expression and then take the square root, the squaring and the square root cancel one another, EXCEPT that the sign will be positive. Thus, if you square 16x, and then take the square root, the answer can be 16x or -16x, depending on whether "x" is positive or negative. Or if you only square x, and then take the square root of the entire expression, it would be 4x or -4x (once again, depending on whether "x" is positive or negative).
x2 actually has two square roots. One of them is indeed x. The other one is negative x.