1. **State the problem:** Solve the quadratic equation $$ax^2 + bx + c = 0$$ for $$x$$. 2. **Rewrite the equation by dividing all terms by $$a$$ (assuming $$a \neq 0$$): $$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$ 3. **Isolate the constant term:** $$x^2 + \frac{b}{a}x = -\frac{c}{a}$$ 4. **Complete the square:** Add $$\left(\frac{b}{2a}\right)^2$$ to both sides to form a perfect square trinomial: $$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$ 5. **Express the left side as a square:** $$\left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}$$ 6. **Simplify the right side by finding a common denominator:** $$\left(x + \frac{b}{2a}\right)^2 = \frac{-4ac + b^2}{4a^2} = \frac{b^2 - 4ac}{4a^2}$$ 7. **Take the square root of both sides:** $$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$ 8. **Solve for $$x$$:** $$x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ **Final answer:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$