Question

Which solution to the equation (3/a+2)+2/a = 4a-4/a^2-4 is extraneous

Answer 1

Solutions : x=4,-2

Extraneous solution : x=-2.

Explanation:

The given equation is

`(3)/(a+2)+(2)/(a)=(4a-4)/(a^2-4)`

Taking LCM we get

`(3a+2(a+2))/((a+2)a)=(4a-4)/((a-2)(a+2))`
`[\because a^2-b^2=(a-b)(a+b)]`

`(3a+2a+4)/((a+2)a)=(4a-4)/((a-2)(a+2))`

Cancel out common factors from the denominators.

`(5a+4)/(a)=(4a-4)/(a-2)`

On cross multiplication we get

`(5a+4)(a-2)=(4a-4)a`

`5a^2-10a+4a-8=4a^2-4a`

`5a^2-6a-8-4a^2+4a=0`

`a^2-2a-8=0`

Splitting the middle term we get

`a^2-4a+2a-8=0`

`a(a-4)+2(a-4)=0`

`(a+2)(a-4)=0`

Using zero product property we get

`a=-2,a=4`

Extraneous solutions: From the solutions of an equation, the invalid solutions are known as extraneous solutions.

For a=-2 right hand side of the given equation is not defined because the denominator become 0.

Therefore, -2 is an extraneous solution.