1. **State the problem:** Solve the equation $X \times X + x\overline{x} = 10$. 2. **Understand the terms:** Here, $X \times X$ means $X^2$. 3. The term $x\overline{x}$ represents the product of a complex number $x$ and its conjugate $\overline{x}$, which equals the magnitude squared: $x\overline{x} = |x|^2$. 4. So the equation becomes: $$X^2 + |x|^2 = 10$$ 5. Without additional information about $X$ and $x$, we can only express the relationship: $$X^2 = 10 - |x|^2$$ 6. This means for any complex number $x$, the value of $X$ satisfies: $$X = \pm \sqrt{10 - |x|^2}$$ 7. Note that $10 - |x|^2 \geq 0$ for $X$ to be real. **Final answer:** $$X = \pm \sqrt{10 - |x|^2}$$