1. **State the problem:** Find the points of intersection between the circle $$x^2 + y^2 = 25$$ and the line $$y = 2x - 12$$. 2. **Formula and rules:** To find intersection points, substitute the expression for $$y$$ from the line into the circle equation and solve for $$x$$. 3. **Substitute:** $$x^2 + (2x - 12)^2 = 25$$ 4. **Expand:** $$x^2 + (4x^2 - 48x + 144) = 25$$ 5. **Combine like terms:** $$x^2 + 4x^2 - 48x + 144 = 25$$ $$5x^2 - 48x + 144 = 25$$ 6. **Bring all terms to one side:** $$5x^2 - 48x + 144 - 25 = 0$$ $$5x^2 - 48x + 119 = 0$$ 7. **Solve quadratic equation:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=5$$, $$b=-48$$, $$c=119$$. Calculate discriminant: $$\Delta = (-48)^2 - 4 \times 5 \times 119 = 2304 - 2380 = -76$$ 8. **Interpret discriminant:** Since $$\Delta < 0$$, there are no real solutions, meaning the line does not intersect the circle. **Final answer:** No real intersection points between the circle $$x^2 + y^2 = 25$$ and the line $$y = 2x - 12$$.