One of the mathematical concepts that produces the most confusion among students relates to the square root operation. Many students mistakenly think that when a term includes a square root operation, both the positive and negative square roots must be taken into account. This is incorrect.
This symbol, √, means "the principal square root" of its operand, by definition. The principal square root is the non-negative number that, when squared, produces the operand. (The same applies to a base with an exponent of ). √ is a function, so it produces a single value always.
However, if, in the course of solving, you want to take the square root of a value, as when you want to eliminate an exponent of 2, you must account for both possible real values for the square root (if the operand isn't zero). So:
Taking the square root of both sides, and explicitly including both the positive and negative square roots:
A digression: The presence of the plus-or-minus sign on the variable term on the left side might surprise you, because you’ve only seen the numeric side of such an equation include the plus-or-minus sign, but there’s nothing special about the numeric side; because both and , when squared, yield , if we want to know all the possibilities for the square root of , we must include both and . However, it turns out that if we enumerate all four possibilities that result from the above equation, we find that there are only two unique equations in the set. Let’s list all the possibilities:
This enumeration produces four equations, but notice that the first two equations are equivalent, and the second two equations are equivalent. The equations in the first pair are both equivalent to
And the equations in the second pair are both equivalent to
Therefore, it is sufficient to write
More generally, this means that when taking the square root of both sides of an equation, we only need to apply the plus-or-minus operation to one side (and you can choose which side earns that privilege, though the numeric side is almost always the best one).
But wait, you might say; if we have to account for both positive and negative roots when taking the square root while solving, doesn't that mean if we have
we have to then write
No -- because we didn't take the square root of both sides of the equation, we merely evaluated the term on the right side (and we wrongly included the negative square root when doing so).
In summary: The square root symbol √ produces only the principal (non-negative) square root of its operand, but when taking the square root of both sides of an equation in the course of solving, you must account for both the positive and negative square roots.