How do you solve this equation? `sqrt (x+3) + sqrt( x-3)=3`  

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lemjay eNotes educator| Certified Educator

`sqrt(x+3) + sqrt(x-3)=3`

To solve, remove the square root. To do so, square both sides.

`(sqrt(x+3) + sqrt(x-3))^2=3^2`

`(sqrt(x+3) + sqrt(x-3))(sqrt(x+3) + sqrt(x-3))=9`

To simplify left side, do FOIL.


`2x + 2sqrt(x+3)sqrt(x-3)=9`

Apply the property of radicals for same indices which is `root(n)(a)*root(n)(c)=root(n)(a*c)` .

`2x + 2sqrt((x+3)(x-3))=9`

Then, to isolate the term with square root, subtract both sides by 2x.

`2x-2x + 2sqrt((x+3)(x-3))=9-2x`


Again, square both sides of the equation to eliminate the square root.



To simplify, do FOIL on both sides of the equation.




Bring together like terms on one side of the equation. So, subtract both sides by `4x^2` .

`4x^2-4x^2 -36=81-36x+4x^2-4x^2`


And subtract both sides by 81.



To isolate x, divide both sides by -36.



Reduce the fraction to its lowest term.


Hence, the solution is `x=13/4` .

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