1. **State the problem:** Solve the equation $x(x-2)=26$ for $x$. 2. **Write the equation:** $$x(x-2)=26$$ 3. **Expand the left side:** $$x^2 - 2x = 26$$ 4. **Bring all terms to one side to set the equation to zero:** $$x^2 - 2x - 26 = 0$$ 5. **Use the quadratic formula:** For $ax^2 + bx + c = 0$, solutions are $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=1$, $b=-2$, $c=-26$. 6. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-2)^2 - 4(1)(-26) = 4 + 104 = 108$$ 7. **Find the square root of the discriminant:** $$\sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3}$$ 8. **Substitute values into the quadratic formula:** $$x = \frac{-(-2) \pm 6\sqrt{3}}{2(1)} = \frac{2 \pm 6\sqrt{3}}{2}$$ 9. **Simplify the expression:** $$x = 1 \pm 3\sqrt{3}$$ **Final answer:** $$x = 1 + 3\sqrt{3} \quad \text{or} \quad x = 1 - 3\sqrt{3}$$