(3x+13)^1/2-5=x. check for extraneous soultions

Both of the solutions, x = {-4, -3}, are viable. There are no extraneous solutions.

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An extraneous solution would show up as the x-intercept associated with the missing branch of the square root function. Since both solutions are on the positive-square-root branch, they are both good solutions to the equation.

For graphing purposes, we have subtracted x to get
   f(x) = √(3x+13) -5 -x
and we are looking for solutions to f(x)=0.

If you add 5 and square both sides, you get the quadratic equation
   3x +13 = (x +5)²
   x² +7x +12 = 0
   (x +3)(x +4) = 0
   x ∈ {-4, -3}

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