Derivation of the Quadratic Formula
Starting from the quadratic equation, we will complete the square to get \(x\) by itself and use the square root method to solve.
$$a x^2 + b x + c = 0$$ $$a x^2 + b x = -c$$Divide through by \(a\)
$$x^2 + \frac{b}{a} x = -\frac{c}{a}$$Complete the square by multiplying \(\frac{b}{a}\) by \(\frac{1}{2}\), squaring the result, and adding it to each side of the equation.
$$x^2 + \frac{b}{a} x + \frac{b^2}{4a^2}= -\frac{c}{a} + \frac{b^2}{4a^2}$$Factoring.
$$\left( x + \frac{b}{2a} \right) ^ 2 = -\frac{c}{a} + \frac{b^2}{4a^2}$$Simplifying right hand side.
$$\left( x + \frac{b}{2a} \right) ^ 2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2}$$ $$\left( x + \frac{b}{2a} \right) ^ 2 = \frac{b^2}{4a^2} - \frac{4ac}{4a^2}$$ $$\left( x + \frac{b}{2a} \right) ^ 2 = \frac{b^2-4ac}{4a^2}$$Take the square root of each side.
$$\sqrt{ \left( x + \frac{b}{2a} \right) ^ 2 } = \pm \sqrt{ \frac{b^2-4ac}{4a^2}}$$Simplifying.
$$x + \frac{b}{2a} = \pm \frac{\sqrt{ b^2-4ac}}{\sqrt{ 4a^2}}$$ $$x + \frac{b}{2a} = \pm \frac{\sqrt{ b^2-4ac}}{2a}$$Getting \(x\) by itself and simplifying.
$$x = - \frac{b}{2a}\pm \frac{\sqrt{ b^2-4ac}}{2a}$$ $$x = \frac{-b \pm \sqrt{ b^2-4ac}}{2a}$$We see that the quadratic formula is found by applying the methods of completing the square and solving by square root to the quadratic equation.