The Quadratic Formula

The formula is completing the square, done once on the general equation instead of on separate numerical examples.

1. Start from $ax^2+bx+c=0$ and divide by $a$.
   $$x^2+\tfrac{b}{a}x+\tfrac{c}{a}=0$$
2. Complete the square on the first two terms.
   $$\left(x+\tfrac{b}{2a}\right)^2-\tfrac{b^2}{4a^2}+\tfrac{c}{a}=0$$
3. Move the numbers across and write the right side as one fraction.
   $$\left(x+\tfrac{b}{2a}\right)^2=\tfrac{b^2-4ac}{4a^2}$$
4. Take the square root of both sides, then subtract $\tfrac{b}{2a}$.
   $$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Strictly, $\sqrt{4a^2}=2|a|$. The $\pm$ already includes both signs, so changing $|a|$ to $a$ does not change the two solutions. The condition $a\ne0$ remains necessary.