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**HallsofIvy** You are right about some things and wrong about others. Given any positive number x, there exist two real numbers whose square is x, one positive and one negative. We write those as $\displaystyle \pm\sqrt{x}$. But the reason we **need** that "$\displaystyle \pm$" is that the symbol, "$\displaystyle \sqrt{x}$" **means** only the positive one. That is, the square root of 4 is *not* "-2 and 2". **The** square root of 4 is 2. The two values of x, such that $\displaystyle x^2= 4$, are -2 and 2. Those are different statements. Given a positive number a, **The** square root of a is $\displaystyle \sqrt{a}$, a positive number. The two values of x, such that $\displaystyle x^2= a$ are $\displaystyle \sqrt{a}$ and $\displaystyle -\sqrt{x}$, which we can write as $\displaystyle \pm\sqrt{a}$. Again, the reason we need that "$\displaystyle \pm$" is that $\displaystyle \sqrt{a}$ alone refers only to the positive root of the equation.