Squaring both sides gives:

`13+sqrt(x)=x+2 sqrt(x) + 1`

Subtract `x+sqrt(x)+1` from both sides:

`12-x=sqrt(x)`

Squaring both sides again gives:

`144-24x+x^2=x`

Subtract `x` from both sides.

`x^2-25x+144=0`

Split `25` into `16` and `9`, which multiply to give `144`.

`x^2-16x-9x+144=0`

`x(x-16)-9(x-16)=0`

`(x-9)(x-6)=0`

This gives us `x = 9` or `x = 16`.

CHECK:

Substituting `x = 9` in our original equation gives:

`"LHS" `

`= sqrt(13+sqrt(9)) `

`= sqrt(13+3)`

`=sqrt(16)`

`=4="RHS" `

Checks OK

Substituting `x = 16` in our original equation gives:

`"LHS" `

`= sqrt(13+sqrt(16)) `

`= sqrt(13+4)`

`=sqrt(17)`

`"RHS" `

`=sqrt(16)+1`

`=4+1=5`

DOES NOT check OK

So we conclude the only solution is `x = 9.`

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