Quadratic Equations General Solution Prooff

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}

So the general solution is $$ x = \frac{-b \pm\sqrt{b^2-4ac}}{2a}$$