## Solution

The quadratic equation $ax^2 + bx + c = 0$ has the solution(s) $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ assuming that $a\neq 0$.

Consider $ax^2 + bx + c = 0$, where $a \neq 0$.

Since $a \neq 0$, we can divide by $a$ to get $x^2 + \frac{b}{a} x + \frac{c}{a} = 0.$

We complete the square.

This shows that the original equation is equivalent to $\left(x + \frac{b}{2a} \right)^2 - \frac{b^2}{4a^2} + \frac{c}{a} = 0.$

Since $x$ appears only once in the equation, we can rearrange this to solve for $x$.

Get the squared term on one side of the equation: $\left(x + \frac{b}{2a} \right)^2 = \frac{b^2}{4a^2} - \frac{c}{a}.$

We can rewrite the right-hand side by putting it over a common denominator: $\left(x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2}.$

We can take the square root of both sides.

Taking account of the possibility of positive and negative square roots, we see $x + \frac{b}{2a} = \frac{\pm\sqrt{b^2 - 4ac}}{2a}.$

Subtracting $\frac{b}{2a}$ from both sides and putting the right-hand side over a common denominator gives $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$