square-root property

The square-root property is the rule for undoing a square. If something squared equals a number, then that something is either the positive or the negative square root of the number. The key idea most beginners forget is that there are two answers, not one — because both a number and its negative square to the same positive value.

Formally: if x^2 = k, then x = sqrt(k) or x = -sqrt(k), often abbreviated x = ±sqrt(k). This works whenever you can isolate a squared expression on one side. It also applies to expressions: if (x - 3)^2 = 16, then x - 3 = ±4, so x = 7 or x = -1.

Two honest cautions. First, taking a square root introduces the ± — drop it and you lose half your solutions. Second, if k is negative there is no real square root; the solutions are then complex numbers. The square-root property is the natural last step after completing the square.