1. **State the problem:** Solve the quadratic equation $$25z^2 - 30z + 4 = -5$$ using the quadratic formula. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero: $$25z^2 - 30z + 4 + 5 = 0$$ $$25z^2 - 30z + 9 = 0$$ 3. **Identify coefficients:** Here, $$a = 25$$, $$b = -30$$, and $$c = 9$$. 4. **Quadratic formula:** The solutions for $$az^2 + bz + c = 0$$ are given by $$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-30)^2 - 4 \times 25 \times 9 = 900 - 900 = 0$$ 6. **Since $$\Delta = 0$$, there is one real repeated root.** 7. **Apply the quadratic formula:** $$z = \frac{-(-30) \pm \sqrt{0}}{2 \times 25} = \frac{30 \pm 0}{50}$$ 8. **Simplify:** $$z = \frac{30}{50} = \frac{3}{5}$$ **Final answer:** $$z = \frac{3}{5}$$