1. **State the problem:** Transform the equation $$2x^2 + 2y^2 + 9x - 4y - 60 = 0$$ into the center-radius form of a circle. 2. **Rewrite the equation:** Divide the entire equation by 2 to simplify coefficients of $x^2$ and $y^2$: $$x^2 + y^2 + \frac{9}{2}x - 2y - 30 = 0$$ 3. **Group $x$ and $y$ terms:** $$x^2 + \frac{9}{2}x + y^2 - 2y = 30$$ 4. **Complete the square for $x$ terms:** Take half of $\frac{9}{2}$ which is $\frac{9}{4}$, square it: $$\left(\frac{9}{4}\right)^2 = \frac{81}{16}$$ 5. **Complete the square for $y$ terms:** Take half of $-2$ which is $-1$, square it: $$(-1)^2 = 1$$ 6. **Add these squares to both sides:** $$x^2 + \frac{9}{2}x + \frac{81}{16} + y^2 - 2y + 1 = 30 + \frac{81}{16} + 1$$ 7. **Rewrite as perfect squares:** $$\left(x + \frac{9}{4}\right)^2 + (y - 1)^2 = 30 + \frac{81}{16} + 1$$ 8. **Simplify the right side:** Convert 30 and 1 to sixteenths: $$30 = \frac{480}{16}, \quad 1 = \frac{16}{16}$$ Sum: $$\frac{480}{16} + \frac{81}{16} + \frac{16}{16} = \frac{577}{16}$$ 9. **Final center-radius form:** $$\left(x + \frac{9}{4}\right)^2 + (y - 1)^2 = \frac{577}{16}$$ This represents a circle with center $$\left(-\frac{9}{4}, 1\right)$$ and radius $$\sqrt{\frac{577}{16}} = \frac{\sqrt{577}}{4}$$.