Consider the general quadratic equation \(ax^2 + bx + c\). We want to find the values of \(x\) that satisfy \(ax^2 + bx + c = 0\). Let’s rearrange things a bit.

\[ax^2 + bx + c = 0 \implies x^2 + \frac{b}{a}x + \frac{c}{a} = 0 \implies (x + \frac{b}{2a})^2 - \frac{b^2}{4a^2} + \frac{c}{a} = 0\] \[\implies (x + \frac{b}{2a})^2 = \frac{b^2}{4a^2} - \frac{c}{a} = \frac{b^2 - 4ac}{4a^2}\] \[\implies x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}\] \[\implies x = \pm \frac{\sqrt{b^2 - 4ac}}{2a} - \frac{b}{2a}\] \[\implies x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]