Extraneous and missing solutions

Extraneous Solution

In mathematics, an extraneous solution represents a solution, such as that to an equation, that is derived using proper procedures, yet results in a mathematical statement that is not true. To understand how this can happen, we will attempt to solve the following equation:

${\frac {1}{x-2}}={\frac {3}{x+2}}-{\frac {6x}{(x-2)(x+2)}}$

To begin solving, we will multiply each side of the equation by the least common denominator of all the fractions contained in the equation. In this case, the LCD is $(x-2)(x+2)$.

After performing these operations, the fractions are eliminated, and the equation becomes:

$x+2=3x-6-6x$

Subtract "3x" from both sides:

$-2x+2=-6-6x$

Add "6x" to both sides:

$4x+2=-6$

Subtract "2" from both sides:

$4x=-8$

Finally, divide each side by "4":

$x=-2$

You can see that we arrive at what appears to be a valid solution rather easily. However, something very strange occurs when we substitute the solution back into the original equation:

${\frac {1}{(-2)-2}}={\frac {3}{(-2)+2}}-{\frac {6(-2)}{((-2)-2)((-2)+2)}}$

The equation then becomes:

${\frac {1}{-4}}={\frac {3}{0}}-{\frac {12}{0}}$

Holy shit!!! The equation cannot be solved!!! The presence of zeros in the denominators is mathematically false. You cannot divide by zero.

This is one reason why solutions to equations must be checked whenever you multiply or divide any side of the equation by an expression containing a variable. On top of that, it must be understood that not all equations will have a solution.