# Quadratic Equations

## General form

$$ax^2 + bx + c = 0 .....(1)$$

## Find general solution

Divide both sides by a
\begin{align}
x^2 + \frac{b}{a}x + \frac{c}{a} & = 0\\\\
x^2 + \frac{b}{a}x & = - \frac{c}{a} .....(2)\\
\end{align}
Now
\begin{align}
\left(x + \frac{b}{2a}\right)^2 &= x^2 + \frac{b}{a}x + \left(\frac{b^2}{4a^2}\right)
\end{align}
Rewrite (2)
\begin{align}
\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2} & = - \frac{c}{a}\\\\
\left(x + \frac{b}{2a}\right)^2 &= \frac{b^2}{4a^2} - \frac{c}{a}\\\\
x + \frac{b}{2a} &= \pm\sqrt{\frac{b^2}{4a^2} - \frac{c}{a}}\\\\
x + \frac{b}{2a} & = \pm\sqrt{\frac{b^2-4ac}{4a^2}}\\\\
x + \frac{b}{2a} & = \frac{\pm\sqrt{b^2-4ac}}{2a}\\\\
x & = -\frac{b}{2a} \frac{\pm\sqrt{b^2-4ac}}{2a}\\\\
x & = \frac{-b \pm\sqrt{b^2-4ac}}{2a}\\\\
\end{align}

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