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

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

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

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

2 x = 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 = x 2

x + 4 = x 2

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

x + 4 = x 2 4 x + 4

0 = x 2 5 x

0 = x ( x 5 )

x = 0   or   x = 5

Check your solutions in the original equation.

Let x = 5 .

5 + 4 = ? 5 2

3 = 3

So, 5 is a solution.

Let x = 0 .

0 + 4 = ? 0 2

2 2

So, 0 is an extraneous solution.


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