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

lemjay | High School Teacher | (Level 3) Senior Educator

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`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.

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

`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`

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

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

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

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

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

`4(x^2-3x+3x-9)=(81-18x-18x+4x^2)`

`4(x^2-9)=81-36x+4x^2`

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

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`

`-36=81-36x`

And subtract both sides by 81.

`-36-81=81-81-36x`

`-117=-36x`

To isolate x, divide both sides by -36.

`(-117)/(-36)=(-36x)/(-36)`

117/36=x

Reduce the fraction to its lowest term.

`13/4=x`

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