1. **State the problem:** Solve the quadratic equation $$x^2 - 25 = 0$$ and sketch its graph. 2. **Formula and rules:** The standard form of a quadratic equation is $$ax^2 + bx + c = 0$$. To solve, use factoring, completing the square, or the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Apply to the equation:** Here, $$a=1$$, $$b=0$$, $$c=-25$$. 4. **Solve by factoring:** $$x^2 - 25 = (x - 5)(x + 5) = 0$$ 5. **Find roots:** Set each factor to zero: $$x - 5 = 0 \Rightarrow x = 5$$ $$x + 5 = 0 \Rightarrow x = -5$$ 6. **Graph explanation:** The parabola opens upward (since $$a=1 > 0$$). The vertex is at $$x=0$$, $$y = -25$$. The roots (x-intercepts) are at $$x = -5$$ and $$x = 5$$. 7. **Final answer:** The solutions are $$x = -5$$ and $$x = 5$$. The graph is a parabola opening upward with vertex at $$(0, -25)$$ and x-intercepts at $$(\pm 5, 0)$$.