Answer
$\{-2\}$.
Work Step by Step
The given expression is
$\Rightarrow \sqrt{x+2}+\sqrt{3x+7}=1$
Subtract $\sqrt{x+2}$ from both sides.
$\Rightarrow \sqrt{x+2}+\sqrt{3x+7}-\sqrt{x+2}=1-\sqrt{x+2}$
Simplify.
$\Rightarrow \sqrt{3x+7}=1-\sqrt{x+2}$
Square both sides.
$\Rightarrow (\sqrt{3x+7})^2=(1-\sqrt{x+2})^2$
Use the special formula $(A-B)^2=A^2-2AB+B^2$
We have $A=1$ and $B=\sqrt{x+2}$
$\Rightarrow 3x+7=(1)^2-2(1)(\sqrt{x+2})+(\sqrt{x+2})^2$
Simplify.
$\Rightarrow 3x+7=1-2\sqrt{x+2}+x+2$
$\Rightarrow 3x+7=-2\sqrt{x+2}+x+3$
Add $-x-3$ to both sides.
$\Rightarrow 3x+7-x-3=-2\sqrt{x+2}+x+3-x-3$
Add like terms.
$\Rightarrow 2x+4=-2\sqrt{x+2}$
Square both sides.
$\Rightarrow (2x+4)^2=(-2\sqrt{x+2})^2$
Use the special formula $(A-B)^2=A^2-2AB+B^2$
We have $A=2x$ and $B=4$
$\Rightarrow (2x)^2+2(2x)(4)+(4)^2=4(x+2)$
Simplify.
$\Rightarrow 4x^2+16x+16=4x+8$
Add $-4x-8$ to both sides.
$\Rightarrow 4x^2+16x+16-4x-8=4x+8-4x-8$
Simplify.
$\Rightarrow 4x^2+12x+8=0$
Factor out $4$ from all terms.
$\Rightarrow 4(x^2+3x+2)=0$
Divide both sides by $4$.
$\Rightarrow \frac{4(x^2+3x+2)}{4}=\frac{0}{4}$
Simplify.
$\Rightarrow x^2+3x+2=0$
Rewrite the middle term $3x$ as $2x+1x$.
$\Rightarrow 0=x^2+2x+1x+2$
Group the terms.
$\Rightarrow 0=(x^2+2x)+(1x+2)$
Factor each group.
$\Rightarrow 0=x(x+2)+1(x+2)$
Factor out $(x+2)$.
$\Rightarrow 0=(x+2)(x+1)$
$\Rightarrow x+2=0$ or $x+1=0$
Isolate $x$.
$\Rightarrow x=-2$ or $x=-1$
Check $x=-2$.
$\Rightarrow \sqrt{-2+2}+\sqrt{3(-2)+7}=1$
$\Rightarrow \sqrt{0}+\sqrt{-6+7}=1$
$\Rightarrow \sqrt{1}=1$
$\Rightarrow 1=1$ True.
Check $x=-1$.
$\Rightarrow \sqrt{-1+2}+\sqrt{3(-1)+7}=1$
$\Rightarrow \sqrt{1}+\sqrt{-3+7}=1$
$\Rightarrow 1+\sqrt{4}=1$
$\Rightarrow 1+2=1$
$\Rightarrow 3=1$ False.
The solution set is $\{-2\}$.
