Answer
It is derived by applying the completion-of-square method on the equation $ ax^{2}+bx+c=0$.
Divide the equation with a, so the leading term is $ x^{2}$
$ x^{2}+(\displaystyle \frac{b}{a})x+(\frac{c}{a})=0$
We rewrite the first two terms as $ x^{2}+2\displaystyle \cdot(\frac{b}{2a})x $
so the missing term to complete a square is $(\displaystyle \frac{b}{2a})^{2}.$
Add it to both sides.
$ x^{2}+(\displaystyle \frac{b}{a})x+(\frac{b}{2a})^{2}+\frac{c}{a}=(\frac{b}{2a})^{2}\quad $... write the perfect square, subtract $\displaystyle \frac{c}{a}$
$(x+\displaystyle \frac{b}{2a})^{2}=(\frac{b}{2a})^{2}-\frac{c}{a}\quad $...simplify the RHS
$(x+\displaystyle \frac{b}{2a})^{2}=\frac{b^{2}-4ac}{4a^{2}}\quad $... apply the square root rule
$ x+\displaystyle \frac{b}{2a}=\pm\sqrt{\frac{b^{2}-4ac}{4a^{2}}}\quad $ ... subtract $\displaystyle \frac{b}{2a}$ and simplify
$ x=\displaystyle \frac{\pm\sqrt{b^{2}-4ac}}{2a}-\frac{b}{2a}$
$ x=\displaystyle \frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\qquad $... (the quadratic formula)
Work Step by Step
Given above.
