The Goal

Find all pairs (x,y) (for x and y both real numbers) satisfying the equation:

The Mistake

Find the algebra mistake:

Need a hint? Look carefully at the red part of the algebra:

The Correction

An Explanation

There is no mistake in the algebra, only a failure to notice at the start that x² + 4y² can only be zero if both x and y are zero.
(Once the algebra had been done, note that the quantity under the square root can only have a real value if x=0, and therefore also y=0.)