Answer
$\color{blue}{\left\{\dfrac{-29}{2} - \dfrac{\sqrt{1241}}{2}, -\dfrac{29}{2}+\dfrac{\sqrt{1241}}{2}\right\}}$
Work Step by Step
Square both sides of the equation:
$$(\sqrt{x}+1)^2=(\sqrt{11-\sqrt{x}})^2
\\(\sqrt{x})^2+2(\sqrt{x})(1)+1^2=11-\sqrt{x}
\\x+2\sqrt{x}+1=11-\sqrt{x}$$
Add $\sqrt{x}$ and subtract $1$ to both sides of the equation:
$$x+2\sqrt{x}+1+\sqrt{x}-1=11-\sqrt{x}+\sqrt{x}-1
\\x+3\sqrt{x}=10$$
Subtract $x$ to both sides of the equation to obtain:
$$3\sqrt{x}=10-x$$
Square both sides of the equation to obtain:
$$(3\sqrt{x})^2=(10-x)^2
\\9(x)=10^2-2(10)(x)+x^2
\\9x=100-20x+x^2$$
Subtract $9x$ to both sides of the equation:
$$9x-9x=100-20x-9x
\\0=100-29x-x^2
\\x^2+29x-100=0$$
The quadratic equation above has $a=1, b=29$, and $c=-100$.
Solve the equation using the quadratic formula to obtain:
$$x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}
\\x=\dfrac{-29\pm \sqrt{29^2-4(1)(-100)}}{2(1)}
\\x=\dfrac{-29 \pm \sqrt{841+400}}{2}
\\x=\dfrac{-29\pm \sqrt{1241}}{2}$$
Thus, the solution set is: $\color{blue}{\left\{\dfrac{-29}{2} - \dfrac{\sqrt{1241}}{2}, -\dfrac{29}{2}+\dfrac{\sqrt{1241}}{2}\right\}}$.
