sat-dog

The correct answer is $-3$. Squaring both sides of the given equation yields ${\left(x-2\right)}^2=3x+34$, which can be rewritten as $x^2-4x+4=3x+34$. Subtracting $3x$ and $34$ from both sides of this equation yields $x^2-7x-30=0$. This quadratic equation can be rewritten as $\left(x-10\right)\left(x+3\right)=0$. According to the zero product property, $\left(x-10\right)\left(x+3\right)$ equals zero when either $x-10=0$ or $x+3=0$. Solving each of these equations for $x$ yields $x=10$ or $x=-3$. Therefore, the given equation has two solutions, $10$ and $-3$. Of these two solutions, $-3$ is the smallest solution to the given equation.