Principal square root
SAT glossary · Math
Definition
The non-negative root denoted by the radical: √(25) = 5, not ± 5. So √(something) = E carries a hidden constraint E ≥ 0; if the non-radical side is a negative constant the answer is "no solution" in three seconds, and it is why y = ±√(25-x^2) needs the ± while the radical alone gives only the upper semicircle.
Principal square root in plain English
The radical symbol sqrtphantomx specifically means the PRINCIPAL (non-negative) square root — so √(25) equals exactly 5, never ±5, even though both 5 and -5 technically square to give 25. This means that any equation of the form √(something)=E has a hidden built-in requirement: E must be zero or positive, since a square root symbol can never spit out a negative number. So if you see an equation where the non-radical side is a negative constant, you can immediately say 'no solution' without doing any actual algebra. This same idea explains why the equation of a full circle, y=±√(25-x^2), NEEDS that plus-or-minus sign tacked on in front — the radical symbol by itself only draws the TOP half of the circle.
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