+ +

Quadratic Formula Derivation

+ +

The standard form of a quadratic equation is:

+ +

a x^{2} + b x + c = 0

+ +

Here, $a, b, c \in ℝ$ , and $a \neq 0$

+ +

We begin by first dividing both sides by the coefficient $a$

+ +

⟹ x^{2} + \frac{b}{a} x + \frac{c}{a} = 0

+ +

We can rearrange the equation:

+ +

x^{2} + \frac{b}{a} x = - \frac{c}{a}

+ +

We can then use the method of completing the square. (Maths is Fun has a really good explanation for this technique)

+ +

x^{2} + \frac{b}{a} x + (\frac{b}{2 a})^{2} = \frac{- c}{a} + (\frac{b}{2 a})^{2}

+ +

On our LHS, we can clearly recognize that it is the expanded form of $(x + d)^{2}$ i.e $x^{2} + 2 x \cdot d + d^{2}$

+ +

⟹ (x + \frac{b}{2 a})^{2} = \frac{- c}{a} + \frac{b^{2}}{4 a^{2}} = \frac{- 4 a c + b^{2}}{4 a^{2}}

+ +

Taking the square root of both sides

+ +

\begin{matrix} x + \frac{b}{2 a} & = \frac{\sqrt{- 4 a c + b^{2}}}{2 a} \\ x & = \frac{\pm \sqrt{- 4 a c + b^{2}} - b}{2 a} \\ = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{matrix}

+ +

This gives you the world famous quadratic formula:

+ +

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

+ +

If you have scrolled this far, consider subscribing to my mailing list here. You can subscribe to either a specific type of post you are interested in, or subscribe to everything with the "Everything" list.

+ +