1. **State the problem:** Solve the quadratic equation $$5x^2 - 6x - 6 = 0$$ by completing the square. 2. **Rewrite the equation:** Move the constant term to the right side: $$5x^2 - 6x = 6$$ 3. **Make the coefficient of $x^2$ equal to 1:** Divide both sides by 5: $$\cancel{5}x^2 - \cancel{5}\frac{6}{5}x = \frac{6}{5}$$ $$x^2 - \frac{6}{5}x = \frac{6}{5}$$ 4. **Complete the square:** Take half of the coefficient of $x$, square it, and add to both sides. Half of $-\frac{6}{5}$ is $-\frac{3}{5}$. Square it: $$\left(-\frac{3}{5}\right)^2 = \frac{9}{25}$$ Add $\frac{9}{25}$ to both sides: $$x^2 - \frac{6}{5}x + \frac{9}{25} = \frac{6}{5} + \frac{9}{25}$$ 5. **Simplify the right side:** $$\frac{6}{5} = \frac{30}{25}$$ So, $$\frac{30}{25} + \frac{9}{25} = \frac{39}{25}$$ 6. **Write the left side as a perfect square:** $$\left(x - \frac{3}{5}\right)^2 = \frac{39}{25}$$ 7. **Solve for $x$ by taking the square root of both sides:** $$x - \frac{3}{5} = \pm \sqrt{\frac{39}{25}} = \pm \frac{\sqrt{39}}{5}$$ 8. **Isolate $x$:** $$x = \frac{3}{5} \pm \frac{\sqrt{39}}{5} = \frac{3 \pm \sqrt{39}}{5}$$ **Final answer:** $$x = \frac{3 \pm \sqrt{39}}{5}$$