Jeremy Nally's Mathematics Tutorials

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.