4. Solve the equation $\quad \sqrt{x+\sqrt{x}-\frac{71}{16}}-\sqrt{x+\sqrt{x}-\frac{87}{16}}=\frac{1}{2}$.
Solution. Taking into account that $x$ is non-negative, we make the substitution $u=\sqrt{x+\sqrt{x}-\frac{71}{16}}, v=\sqrt{x+\sqrt{x}-\frac{87}{16}}, u \geq 0, v \geq 0$.

Then we obtain the system $\left\{\begin{array}{l}u-v=1 / 2, \\ u^{2}-v^{2}=1,\end{array} \Rightarrow\left\{\begin{array}{l}u-v=1 / 2, \\ u+v=2,\end{array} \Rightarrow\left\{\begin{array}{l}u=5 / 4, \\ v=3 / 4,\end{array} \Rightarrow\right.\right.\right.$ reverse substitution: $\left\{\begin{array}{l}x+\sqrt{x}-\frac{71}{16}=\frac{25}{16}, \\ x+\sqrt{x}-\frac{87}{16}=\frac{9}{16} .\end{array} \Rightarrow x+\sqrt{x}-6=0 \Rightarrow x=4\right.$.

## Answer: 4.