To solve the quadratic equation $ax^2 + bx + c = 0$ by completing the square, follow these steps:

Steps:

1. Divide the equation by $a$ (if $a \neq 1$) to make the coefficient of $x^2$ equal to 1: $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$

2. Move the constant term $\frac{c}{a}$ to the other side: $x^2 + \frac{b}{a}x = -\frac{c}{a}$

3. Complete the square on the left-hand side. To do this, take half of the coefficient of $x$, square it, and add it to both sides: $\text{Half of } \frac{b}{a} = \frac{b}{2a}, \quad \left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2}$ Add $\frac{b^2}{4a^2}$ to both sides: $x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}$

4. Simplify both sides: The left-hand side is now a perfect square: $\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$

5. Take the square root of both sides: $x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$

6. Solve for $x$ by isolating it: $x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}$

Final solution:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This is the quadratic formula, derived through completing the square.

Would you like more details on any step, or have questions?

Here are 5 related questions to explore further:

1. Can you apply this method to solve $3x^2 + 6x - 9 = 0$?
2. What happens if the discriminant $b^2 - 4ac$ is negative?
3. How does completing the square relate to the vertex form of a quadratic function?
4. Can you complete the square for non-monic quadratics, where $a \neq 1$?
5. How does completing the square help in deriving the standard form of a circle?

Tip: Completing the square is also useful for transforming quadratics into vertex form, which reveals key properties like the vertex and axis of symmetry.