Extraneous Roots
Because equations involving rational expressions have variables in denominators, a
root to the equation might cause a 0 to appear in a denominator. In this case the root
does not satisfy the original equation, and so it is called an extraneous root.
Example 1
An equation with an extraneous root
Solve
Solution
Because x^{2}  2x = x(x  2), the LCD for x, x  2, and x^{2}
 2x is x(x  2).


Multiply each side by x(x  2). 
3(x  2) + 6x 3x  6 + 6x
9x  6
9x
x 
= 12
= 12
= 12
= 18
= 2 

Neither 0 nor 2 could be a solution because replacing x by either 0 or 2 would cause
0 to appear in a denominator in the original equation. So 2 is an extraneous root and
the solution set is the empty set, Ã˜.
Example 2
An equation with an extraneous root
Solve
Solution
Because the LCD is x  2, we multiply each side by x  2:


x^{2}  4 +
x 
= 2 
x^{2} + x 
6 
= 0 
(x + 3)(x  2) 
= 0 
x + 3 = 0 
or 
x  2 
= 0 
x = 3 
or 
x 
= 2 
Replacing x by 2 in the original equation would cause 0 to appear in a denominator.
So 2 is an extraneous root. Check that the original equation is satisfied if x =
3. The solution set is {3}.
