1. Let's start by stating the problem: solve the quadratic equation by completing the square. 2. Consider a quadratic equation in the form $$ax^2 + bx + c = 0$$ where $$a \neq 0$$. 3. To solve by completing the square, first divide the entire equation by $$a$$ (if $$a \neq 1$$) to simplify it to $$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$. 4. Move the constant term to the right side: $$x^2 + \frac{b}{a}x = -\frac{c}{a}$$. 5. Add the square of half the coefficient of $$x$$ to both sides to complete the square: add $$\left(\frac{b}{2a}\right)^2$$ to both sides. 6. This gives: $$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$. 7. The left side can be written as a perfect square: $$\left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}$$. 8. Simplify the right side: $$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$$. 9. Take the square root of both sides: $$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$. 10. Isolate $$x$$: $$x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$. 11. Combine into the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 12. **In summary**, completing the square transforms the quadratic into a perfect square trinomial to find solutions for $$x$$. This method not only helps solve quadratics but also derives the quadratic formula.