1. **State the problem:** Solve the equation $$x - 5 = 2\sqrt{x} - 5$$ for $x$. 2. **Isolate the square root term:** Add 5 to both sides to simplify: $$x - 5 + 5 = 2\sqrt{x} - 5 + 5$$ $$x = 2\sqrt{x}$$ 3. **Square both sides to eliminate the square root:** $$x^2 = (2\sqrt{x})^2$$ $$x^2 = 4x$$ 4. **Rewrite the equation:** $$x^2 - 4x = 0$$ 5. **Factor the equation:** $$x(x - 4) = 0$$ 6. **Solve for $x$:** Set each factor equal to zero: - $x = 0$ - $x - 4 = 0 \Rightarrow x = 4$ 7. **Check for extraneous solutions:** Substitute $x=0$ into the original equation: $$0 - 5 = 2\sqrt{0} - 5$$ $$-5 = 0 - 5$$ $$-5 = -5$$ (True) Substitute $x=4$ into the original equation: $$4 - 5 = 2\sqrt{4} - 5$$ $$-1 = 2 \times 2 - 5$$ $$-1 = 4 - 5$$ $$-1 = -1$$ (True) **Both solutions are valid.** **Final answer:** $$x = 0 \text{ or } x = 4$$