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 non-negative operand. (The same applies to a base with an exponent of 
). √ is a function, so it produces a single value always.
Therefore:
It doesn’t matter what your friends, your parents, or even your math teachers have told you:
The square root operator √ and the equivalent exponent only produce one value, and that one value is the non-negative root.
And just to address some misinformation that’s circulating about this fact, this is not some sort of rule that is special to the SAT only, and it is most assuredly not a “US math thing;” this definition applies everywhere on the planet, and, we have it on good authority, in outer space as well.
However…
If you want to eliminate an exponent of 2 on a term when manipulating the term algebraically as when you’re solving for a variable, you must account for both possible real values of the square root. For example, here we have an equation with on one side, and we’d like to know the value(s) of that satisfy the equation:
We want to eliminate the exponent of 2 on so we end up with an equation (or equations) of the form [some value] that tells us the solution(s). We can do that by taking the square roots of both sides of the equation. Notice that we said square roots, plural, because both the positive and the negative roots would satisfy the equation. Remember, the familiar symbol √ produces only the principal (non-negative) square root, and, oddly, there is no single mathematical symbol that indicates both square roots are to be produced, so we made one up:
We can apply this operation to both sides of our equation:
Keeping in mind that there are two roots of , which are and , and two roots of 4, which are 2 and -2, we can take the square roots of both sides and explicitly include both the positive and negative roots by prefixing the terms with the plus-or-minus symbol ±:
Warning: Geeky math discussion ahead. Obviously, +2 is the positive (principal) square root of 4, and -2 is the negative root. However, we don’t know whether or is the principal square root of , because we don’t know whether the value of is positive or negative. All we know is that is one root, and is the other (more about finding the principal square root of an expression like a bit later). What matters here, though, is that we have both roots accounted for.
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 , which we’ve notated as 
, we must include both and . However, it turns out that if we enumerate all four combinations 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 roots 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 choice).
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
The answer is no — because we didn’t take the square roots 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).
If you’re still here, you’re about to be sorry! We’re going to look at this from two more angles.
You might have wondered why we bothered to devise our own mathematical operator meaning “both square roots” instead of simply combining the usual square root operator with the plus-minus operator ± and writing
The way the right side of this equation produces the two roots is straightforward. First, we apply the square root operator (which, remember, produces just the non-negative root):
Next, we apply the plus-minus operator to produce two values:
Simple enough. Now, let’s process the operations on the left side. First, we apply the square root operator:
This is the tricky part. You might think that the square root of is obviously just — but you’d be wrong. Recall that the square root operator returns the principal square root, and that means the non-negative square root of the operand. Here, we do not know the value of , so we don’t know if the principal square root of is or .
How do we resolve this? We’ll have to apply another operation that changes the sign of a value from negative to positive if the value is negative; that operation is absolute value. Here’s how that looks:
Now, we can continue with our processing of this term by applying the plus-minus operator and getting the two values for the square roots:
Where are we now? We have two values for the left side of our equation, and two values for the right side; because we know the sign of each of those values, we can match them up:
Well, this isn’t very interesting — the second equation is just the first one multiplied by -1, so we don’t need it at all; what we’re left with is this:
We can now solve for at last by breaking out the two cases that result from the absolute value operation:
We’ve ended up with the same solutions. Now, was it necessary to go through the intermediate steps involving the absolute value operation? What if we just employed this sequence of steps:
This does produce the correct set of solutions, but it obscures the fact that we don’t actually know if or if instead because it misleadingly indicates that (which is not a valid equation), even though we arrive at the correct pair of solutions in the end, so we like the idea of avoiding that complication by using a single operator that returns both square roots, though we don’t know which one is the principal root.
We don’t want to overlook a surprising math fact that we demonstrated above:
As explained above, the reason for this is that the square root operator √ produces one value only, and that value is the principal (non-negative) square root of its operand, but when the operand is a variable, unless otherwise specified, we do not know whether the operand is positive or negative. For example,
Anyone who has read this far deserved more of this square root zaniness. Let’s look at another approach to eliminating an exponent of 2 on a variable while solving: we’ll first find the principal square root of both sides:
We see here that the absolute value of is 2; as you probably know, an equation with an absolute value operation on one side really represents two different equations: one equation where the expression that is the operand of the absolute value operation is positive, and a second equation where that operand is negative. As we showed earlier, breaking out those two equations gives us our solution:
If you’re reading this, you’re clearly a glutton for punishment, so we will look at this situation through one more lens: Quadratics!
You might not have realized it due to the structure of the equation, but is actually a quadratic equation, and a special case of one at that. Let’s do some rearrangement to make that clearer:
This is just a simple quadratic equation in standard form , where . But the quadratic expression on the left side has another important attribute: it represents a difference of squares. To refresh your memory, here’s the pattern for a difference of squares expression and the most useful way to factor it:
In our quadratic expression , can be and can be 2, so the pattern is represented by . Applying the difference of squares factoring principle, we can factor thusly:
This should look very familiar! At this point, we know that when either factor equals zero, the equation is valid, so the roots are and , or . Notice that we’ve now connected the term “roots” in “square roots” with that term’s meaning in the context of the solutions of an equation; you might not have realized this connection previously.
If you’re sad that this treatise is drawing to a close, we’ll point out one more square-root-related fact. You might have noticed that we said that when matching our quadratic expression with the difference of squares pattern , can be and can be 2. Why didn’t we simply say that is and is 2? The answer is that we can use either square root of each of those terms, because either root, when squared, produces the same result, by definition. Therefore, instead of choosing for , we could have chosen , because , and instead of choosing 2 for , we could have chosen -2, because . Does the difference of squares factorization still hold if we make these alternative choices? Yes; like honey badger, that transformation don’t care. If you’d like to reassure yourself that this is the case, go ahead and perform these alternative substitutions to produce a factored expression, and then perform the distribution to reverse the factoring process, and see where you end up. However, when choosing the terms to use in a difference of squares factorization, we recommend sticking with the principal square root for numeric values, so representing as rather than as , and using the provided base (or what we cheekily call the “symbolic principal square root”) for variables, thus representing as, well, , not as .
Congratulations! You’ve made it to the finish line in the square root event of the SAT math geek Olympics. We will wind up with this summary, in case the main point has become gauzy in the pursuit of completeness: The square root symbol √ produces only the principal (non-negative) square root of its operand, but when taking the square roots of both sides of an equation to eliminate an exponent of 2 in the course of solving for a variable, you must account for both the positive and negative square roots.
