1. **State the problem:** Solve the equation $$\sqrt{x-1} = x - 5$$ for $x$. 2. **Recall the formula and rules:** To solve equations involving square roots, we isolate the square root and then square both sides to eliminate it. Remember, the expression inside the square root must be non-negative: $$x - 1 \geq 0 \Rightarrow x \geq 1$$. 3. **Square both sides:** $$\left(\sqrt{x-1}\right)^2 = (x - 5)^2$$ $$x - 1 = (x - 5)^2$$ 4. **Expand the right side:** $$x - 1 = (x - 5)(x - 5) = x^2 - 10x + 25$$ 5. **Bring all terms to one side:** $$0 = x^2 - 10x + 25 - x + 1$$ $$0 = x^2 - 11x + 26$$ 6. **Solve the quadratic equation:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-11$, $c=26$. Calculate the discriminant: $$\Delta = (-11)^2 - 4 \times 1 \times 26 = 121 - 104 = 17$$ Calculate the roots: $$x = \frac{11 \pm \sqrt{17}}{2}$$ 7. **Check for extraneous solutions:** Recall the domain $x \geq 1$ and the original equation. - For $$x = \frac{11 + \sqrt{17}}{2} \approx 8.56$$: $$\sqrt{8.56 - 1} \approx \sqrt{7.56} \approx 2.75$$ $$8.56 - 5 = 3.56$$ Not equal, so discard. - For $$x = \frac{11 - \sqrt{17}}{2} \approx 2.44$$: $$\sqrt{2.44 - 1} = \sqrt{1.44} = 1.2$$ $$2.44 - 5 = -2.56$$ Not equal, discard. 8. **Check the original equation carefully:** Since the right side is $x-5$, it must be non-negative (because the left side is a square root and always non-negative). So: $$x - 5 \geq 0 \Rightarrow x \geq 5$$ Reconsider the roots with this domain: - $$x \approx 8.56$$ satisfies $x \geq 5$. - $$x \approx 2.44$$ does not. Check $x \approx 8.56$ again: $$\sqrt{8.56 - 1} = \sqrt{7.56} \approx 2.75$$ $$8.56 - 5 = 3.56$$ Not equal, so no solution. 9. **Conclusion:** No real $x$ satisfies the equation $$\sqrt{x-1} = x - 5$$. **Final answer:** No solution in real numbers.