1. **State the problem:** Solve the quadratic equation $$4x^2 - 5 = 4(x - 5)$$. 2. **Rewrite the equation:** Expand the right side: $$4x^2 - 5 = 4x - 20$$ 3. **Bring all terms to one side:** $$4x^2 - 5 - 4x + 20 = 0$$ Simplify: $$4x^2 - 4x + 15 = 0$$ 4. **Identify coefficients:** Here, $$a = 4$$, $$b = -4$$, and $$c = 15$$. 5. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 6. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-4)^2 - 4 \times 4 \times 15 = 16 - 240 = -224$$ 7. **Interpret the discriminant:** Since $$\Delta < 0$$, there are no real solutions; the solutions are complex. 8. **Find the complex solutions:** $$x = \frac{-(-4) \pm \sqrt{-224}}{2 \times 4} = \frac{4 \pm \sqrt{-224}}{8}$$ 9. **Simplify the square root:** $$\sqrt{-224} = \sqrt{-1 \times 224} = i \sqrt{224}$$ Factor 224: $$224 = 16 \times 14$$ So, $$\sqrt{224} = \sqrt{16 \times 14} = 4 \sqrt{14}$$ 10. **Write the final solutions:** $$x = \frac{4 \pm 4i \sqrt{14}}{8} = \frac{4}{8} \pm \frac{4i \sqrt{14}}{8} = \frac{1}{2} \pm \frac{i \sqrt{14}}{2}$$ **Final answer:** $$x = \frac{1}{2} \pm \frac{i \sqrt{14}}{2}$$