Extraneous Solutions

An extraneous solution is a root of a transformed equation that is not a root of the original equation because it was excluded from the domain of the original equation.

Example 1:

Solve for x , 1 x2 + 1 x+2 = 4 ( x2 )( x+2 ) .

1 x2 + 1 x+2 = 4 ( x2 )( x+2 )

( x2 )( x+2 ) ( x2 ) + ( x2 )( x+2 ) ( x+2 ) = 4( x2 )( x+2 ) ( x2 )( x+2 )

( x2 )+( x+2 )=4

2x=4

x=2

But 2 is excluded from the domain of the original equation because it would make the denominator of one of the fractions zero--and division by zero is not allowed!  .  Therefore, it cannot be a root of the original equation.  So, 2 is an extraneous solution. So, the equation has no solutions.


Example 2:

Solve for x , x+4 =x2

x+4 =x2

( x+4 ) 2 = ( x2 ) 2

x+4= x 2 4x+4

0= x 2 5x

0=x( x5 )

x=0  or  x=5

Check your solutions in the original equation.

Let x=5 .

5+4 = ? 52

3=3

So, 5 is a solution.

Let x=0 .

0+4 = ? 02

22

So, 0 is an extraneous solution.