Question

How do you solve `(2a-5)/(a-9)+a/(a+9)=-6/(a^2-81)`?

Answer 1

Solution : `a=3 or a =-13/3`

Explanation:

`(2a-5)/(a-9) +a /(a+9) = - 6/((a^2-81)` or

`(2a-5)/(a-9) +a /(a+9) = - 6/((a+9(a-9)` Multiplying by `(a+9(a-9)` on boh sides we get,

`(2a-5)(a+9) +a(a-9) = - 6` or

`2a^2 +13a -45 +a^2 -9a +6 =0` or

`3a^2+4a-39=0 or 3a^2+13a-9a-39=0` or

`a(3a+13) -3(3a+13) =0` or

`(3a+13)(a-3)=0`, either `3a+13=0 :. a = -13/3` or

`a-3=0:.a =3`

Solution: `a=3 or a =-13/3` [Ans]

Answer 2

No real solutions and `x=-13/3,x=3` extraneous solutions

Explanation:

`(2a-5)/(a-9)+(a)/(a+9)=- 6/(a^2-81)`

`(2a-5)/(a-9)+(a)/(a+9)+ 6/((a-9)(a+9))=0`

`((a+9)(2a-5)+a(a-9)+6(a+9))/((a-9)(a+9))=0`

Multiply both sides by `(a-9)(a+9)`

`2a^2+13a-45+a^2-9a+6a+54=0`

`3a^2+10a+9=0`

`a=(-b+-sqrt(b^2-4ac))/(2a)`

`a=(-(10)+-sqrt((10)^2-4(3)(9)))/(2(3))`

`a=(-10+-sqrt(100-108))/(2(3))`

`a=(-10+-sqrt(-8))/6`

No real solution because of `sqrt(-8)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Alternate method:

`(2a-5)/(a-9)+(a)/(a+9)=- 6/(a^2-81)`

`(2a-5)/(a-9)+(a)/(a+9)=- 6/((a-9)(a+9))`

`((a+9)(2a-5)+(a-9)a=-6)/((a-9)(a+9))`

Multiply both sides by `(a-9)(a+9)`

`(a+9)(2a-5)+(a-9)a=-6`

`2a^2+13a-45+a^2-9a=-6`

`3a^2+4a-39=0`

`(3a+13)(a-3)=0`

`3a=-13 or a=3`

`a=-13/3 or a=3`

extraneous solutions