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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 x2 - 2x = x(x - 2), the LCD for x, x - 2, and x2 - 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, Ø.

 

Example 2

An equation with an extraneous root

Solve

Solution

Because the LCD is x - 2, we multiply each side by x - 2:

x2 - 4 + x = 2
x2 + 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}.

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