When an equation has a square root or a variable in a denominator, solving can produce values that do not actually work in the original equation. These are called extraneous solutions. The SAT writes questions specifically to catch students who solve correctly and never check.
Here is where they come from. Squaring both sides of an equation is not fully reversible, because two different numbers can have the same square. The statement 3 equals negative 3 is false, but squaring both sides gives 9 equals 9, which is true. So squaring can turn a false statement into a true one, which means it can introduce solutions that were never valid.
Work an example. Solve the square root of the quantity x plus 6 equals x. Square both sides to get x plus 6 equals x squared, rearrange to x squared minus x minus 6 equals zero, and factor into the quantity x minus 3 times the quantity x plus 2 equals zero. The candidates are x equals 3 and x equals negative 2.
Now test both in the original. For x equals 3, the left side is the square root of 9, which is 3, and the right side is 3, so it works. For x equals negative 2, the left side is the square root of 4, which is 2, and the right side is negative 2. Since 2 does not equal negative 2, that candidate is extraneous and the only solution is 3. The reason is structural: the square root symbol returns the nonnegative root, so it can never equal a negative number.
Rational equations have the same problem for a different reason. If you solve 1 over the quantity x minus 5 equals 3 over the quantity x squared minus 25 and arrive at x equals 5, that value makes the original denominators zero, so the expression is undefined there and the solution must be discarded. Before you finish, list the values that would make any denominator zero and cross off any candidate that matches.
The practical habit is short. Solve as usual, then substitute every candidate back into the original equation, not into a line you produced along the way. Checking against an intermediate equation defeats the purpose, since the intermediate equation is exactly the one that gained the fake solution.